Measurement of the Proton Recoil
Spectrum in Neutron Beta Decay
with the Spectrometer aSPECT:
Study of Systematic Effects
Dissertation
zur Erlangung des Grades
“Doktor der Naturwissenschaften”
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universität
in Mainz
vorgelegt von
Dipl.-Math. Gertrud Emilie Konrad
geboren in Worms
Mainz, im August 2011
Dean:
First referee:
Second referee:
Oral examination: 24 January, 2012
D77
ii
Properly speaking, such work is never finished;
one must declare it so when,
according to time and circumstances,
one has done one’s best.
J. W. von Goethe, Italian Journey
Caserta, 16 March 1787
iii
iv
Abstract
Free neutron decay, n→ peνe, is the simplest nuclear beta decay, well described as a purely
left-handed, vector minus axial-vector interaction within the framework of the Standard
Model (SM) of elementary particles and fields. Due to its highly precise theoretical descrip-
tion, neutron beta decay data can be used to test certain extensions to the SM. Possible
extensions require, e.g., new symmetry concepts like left-right symmetry, new particles,
leptoquarks, supersymmetry, or the like. Precision measurements of observables in neutron
beta decay address important open questions of particle physics and cosmology, and are
generally complementary to direct searches for new physics beyond the SM in high-energy
physics.
In this doctoral thesis, a measurement of the proton recoil spectrum with the neutron
decay spectrometer aSPECT is described. From the proton spectrum the antineutrino-
electron angular correlation coefficient a can be derived. In our first beam time at the
Forschungs-Neutronenquelle Heinz Maier-Leibnitz in Munich, Germany (2005-2006), back-
ground instabilities due to particle trapping and the electronic noise level of the proton
detector prevented us from presenting a new value for a. In the latest beam time at the
Institut Laue-Langevin (ILL) in Grenoble, France (2007-2008), the trapped particle back-
ground has been reduced sufficiently and the electronic noise problem has essentially been
solved. For the first time, a silicon drift detector was used. As a result of the data analysis,
we identified and fixed a problem in the detector electronics which caused a significant
systematic error. The target figure of the latest beam time was a new value for a with a
total relative error well below the present literature value of 4%. A statistical accuracy
of about 1.4% was reached, but we could only set upper limits on the correction of the
problem in the detector electronics, which are too high to determine a meaningful result.
The present doctoral thesis focused on the investigation of several different systematic
effects. With the knowledge of the systematic effects gained in this thesis, we are now able
to improve the aSPECT spectrometer to perform a 1% measurement of a in a further
beam time at the ILL.
Zusammenfassung
Der Zerfall des freien Neutrons, n → peνe, ist der einfachste Kernbetazerfall. Die-
ser wird im Rahmen des Standardmodells (SM) der Elementarteilchenphysik gut als eine
rein linkshändige Vektor-Minus-Axialvektor-Wechselwirkung beschrieben. Aufgrund seiner
äußerst präzisen theoretischen Beschreibung können Neutronenbetazerfallsdaten für Tests
bestimmter Erweiterungen des SMs verwendet werden. Mögliche Erweiterungen erfordern,
z.B., neue Symmetriekonzepte wie Links-Rechts-Symmetrie, neue Teilchen, Leptoquarks,
Supersymmetrie, oder ähnliches. Präzisionsmessungen von Observablen im Neutronenbe-
tazerfall beschäftigen sich mit wichtigen offenen Fragen der Teilchenphysik und Kosmologie
und sind im Allgemeinen komplementär zur direkten Suche nach neuer Physik jenseits des
SMs in der Hochenergiephysik.
In dieser Doktorarbeit wird eine Messung des Protonenrückstoßspektrums mit dem
Neutronenzerfallsspektrometer aSPECT beschrieben. Aus dem Protonenspektrum lässt
sich der Antineutrino-Elektron Winkelkorrelationskoeffizient a ableiten. Bei unserer ersten
Strahlzeit an der Forschungs-Neutronenquelle Heinz Maier-Leibnitz in München, Deutsch-
land (2005-2006), haben uns Untergrundinstabilitäten, verursacht durch gespeicherte Teil-
chen und den elektronischen Rauschpegel des Protonendetektors, daran gehindert, einen
neuen Wert für a zu präsentieren. Bei der letzten Strahlzeit am Institut Laue-Langevin
(ILL) in Grenoble, Frankreich (2007-2008), wurde der Untergrund aufgrund von gespei-
cherten Teilchen ausreichend reduziert und das Problem des elektronischen Rauschens
im Wesentlichen gelöst. Zum ersten Mal wurde ein Siliziumdriftdetektor verwendet. Als
Ergebnis der Datenanalyse haben wir ein Problem in der Detektorelektronik identifiziert
und behoben, welches einen signifikanten systematischen Fehler verursacht hat. Die Ziel-
größe der letzten Strahlzeit war ein neuer Wert für a mit einem relativen Gesamtfehler
von weniger als dem gegenwärtigen Literaturwert von 4%. Zwar wurde eine statistische
Genauigkeit von rund 1.4% erreicht, allerdings konnten wir für die Korrektur des Pro-
blems in der Detektorelektronik nur Obergrenzen festlegen, welche zu hoch sind, um ein
aussagekräftiges Ergebnis zu bestimmen. Im Mittelpunkt der vorliegenden Doktorarbeit
stand die Untersuchung von mehreren verschiedenen systematischen Effekten. Mit den Er-
kenntnissen über die systematischen Effekte, die in dieser Arbeit gewonnen wurden, sind
wir in der Lage, das aSPECT Spektrometer zu verbessern, um eine 1% Messung von a in
einer weiteren Strahlzeit am ILL durchzuführen.
Publications
Some ideas and figures presented in this thesis have appeared previously in the following
publications:
• G. Konrad et al., Design of an Anti-Magnetic Screen for the Neutron Decay Spec-
trometer aSPECT, in Proceedings of the European Comsol Conference, Grenoble,
France, 2007, p. 241-245 (2007) [1]
• Report for the TRAPSPEC JRA-11 in EURONS on task T-J11-1 (Simulations and
calculations), 2008 [2]
• Report for the TRAPSPEC JRA-11 in EURONS on task T-J11-8 (Neutron decay
retardation spectrometer), 2008 [3]
• Measurement of the Proton Spectrum in free Neutron Decay, ILL Experimental
Report, 2009 [4]
• G. Konrad et al., The Proton Spectrum in Neutron Beta Decay: Latest Results with
the aSPECT Spectrometer, in Proceedings of the eighteenth Particles and Nuclei
International Conference, Eilat, Israel, 2008, Nucl. Phys. A 827, 529c-531c (2009)
[5]
• G. Konrad et al., Impact of Neutron Decay Experiments on non-Standard Model
Physics, in Proceedings of the 5th International BEYOND 2010 Conference, Cape
Town, South Africa, 2010, World Scientific, Singapore, ISBN 978-981-4340-85-4,
p. 660-672, 2011, and arXiv:1007.3027v2 (2010) [6]
• G. Konrad et al., Neutron Decay with PERC: a Progress Report, in Proceedings of
the 5th European Conference on Neutron Scattering, Prague, Czech Republic, 2011,
J. Phys.: Conf. Series, accepted (2011) [7]
vii
viii
Contents
Abstract i
Publications vii
List of Figures xiii
List of Tables xvii
1 Introduction 1
2 Neutron Beta Decay 5
2.1 Theory of Neutron Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Weak Interaction in the SM . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Neutron Beta Decay in the SM . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Measurable Parameters in Neutron Beta Decay . . . . . . . . . . . 10
2.1.5 The Proton and Lepton Spectra . . . . . . . . . . . . . . . . . . . . 13
2.1.6 Extensions of the Standard Model . . . . . . . . . . . . . . . . . . 19
2.2 Tests of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Test of the V−A Description . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Unitarity of the CKM Matrix and Determination of |Vud| and λ . . 25
2.3 Searches for Physics Beyond the Standard Model . . . . . . . . . . . . . . 26
2.3.1 Constraints from Neutron Decay Alone . . . . . . . . . . . . . . . . 27
2.3.2 Left-Handed Coupling Constraints from a Combined Analysis
of Neutron and Nuclear Decays . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Limits from Other Fields . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Previous and Competing Measurements of a . . . . . . . . . . . . . . . . . 35
2.4.1 Previous Measurements . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.2 Upcoming Experiments . . . . . . . . . . . . . . . . . . . . . . . . 39
3 The aSPECT Experiment 43
3.1 Measurement Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Adiabatic Invariance and Magnetic Mirror Effect . . . . . . . . . . 44
3.1.2 The Adiabatic Transmission Function . . . . . . . . . . . . . . . . 47
3.1.3 Computation of a from the Proton Recoil Spectrum . . . . . . . . 50
3.2 The Retardation Spectrometer aSPECT . . . . . . . . . . . . . . . . . . . 53
3.2.1 The Electrode System . . . . . . . . . . . . . . . . . . . . . . . . . 53
ix
x CONTENTS
3.2.2 The Superconducting Coil System . . . . . . . . . . . . . . . . . . 62
3.3 The Detection System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.1 The Proton Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3.2 Signal Processing Electronics . . . . . . . . . . . . . . . . . . . . . 68
3.4 Systematic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.1 The Adiabatic Transmission Function . . . . . . . . . . . . . . . . 71
3.4.2 Non-Adiabatic Proton Motion . . . . . . . . . . . . . . . . . . . . . 72
3.4.3 Residual Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.5 Doppler Effect due to Neutron Motion . . . . . . . . . . . . . . . . 76
3.4.6 Edge Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.7 Detection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Measurements at the ILL 81
4.1 Experimental Set-up at the ILL . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.1 The Neutron Beam Facility PF1b of the ILL . . . . . . . . . . . . 82
4.1.2 The Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.3 Neutron Beam Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1.4 Magnetic Field Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 Neutron Beam Monitor . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Monitoring of the Barrier Potential . . . . . . . . . . . . . . . . . . 90
4.2.3 Energy Calibration of the Proton Detector . . . . . . . . . . . . . . 91
4.2.4 Measurement Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2.5 Investigation of Systematic Effects . . . . . . . . . . . . . . . . . . 93
5 Data Analysis 95
5.1 Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.1 The Fit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.1.2 Fit States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.3 Refitted Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Extraction of a from the Proton Spectra . . . . . . . . . . . . . . . . . . . 99
5.2.1 Dead Time Correction . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 Background Correction . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.3 Integration of the Count Rate . . . . . . . . . . . . . . . . . . . . . 106
5.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.1 Dependence on the Barrier Potential . . . . . . . . . . . . . . . . . 106
5.3.2 Dependence on the Lower Dipole Potential . . . . . . . . . . . . . 110
5.3.3 Trapping of Decay Protons . . . . . . . . . . . . . . . . . . . . . . 112
5.3.4 Trapping of Decay Electrons . . . . . . . . . . . . . . . . . . . . . . 113
5.4 Unexpected Systematic Effects . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4.1 Dependence of a on the Lower Integration Limit . . . . . . . . . . 115
5.4.2 Backscattering of Decay Electrons Inside the Spectrometer . . . . . 128
5.4.3 Dependence of a on the Electrostatic Mirror Potential . . . . . . . 137
5.4.4 Charging of the Collimation System . . . . . . . . . . . . . . . . . 140
5.5 Investigations of Systematic Effects . . . . . . . . . . . . . . . . . . . . . . 143
5.5.1 Electrostatic Mirror Potential . . . . . . . . . . . . . . . . . . . . . 143
5.5.2 Magnetic Mirror Effect in the Decay Volume . . . . . . . . . . . . 144
CONTENTS xi
5.5.3 Magnetic Mirror Effect in Front of the Proton Detector . . . . . . 147
5.5.4 Ratio of the Magnetic Fields . . . . . . . . . . . . . . . . . . . . . 148
5.5.5 Height of the Main Magnetic Field . . . . . . . . . . . . . . . . . . 149
5.5.6 Edge Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 The Antineutrino-Electron Correlation Coefficient a . . . . . . . . . . . . 164
5.6.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.6.2 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Investigations of the Patch Effect 167
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.1 Surface Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2 Operation of the Kelvin Probe . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.2 The “off-null” Technique . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2.3 Work Function Measurement . . . . . . . . . . . . . . . . . . . . . 173
6.2.4 Reference Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3 Samples, Data, and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3.1 Details of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3.2 Kelvin Probe Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3.3 Gradient Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3.4 Temporal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.3.5 Reproducibility of the Samples . . . . . . . . . . . . . . . . . . . . 183
6.3.6 Surface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.3.7 Sputtered Gold and SEM Analysis . . . . . . . . . . . . . . . . . . 190
6.4 Influence of the Patch Effect on a . . . . . . . . . . . . . . . . . . . . . . . 192
6.4.1 Patch Effect in the Analyzing Plane . . . . . . . . . . . . . . . . . 192
6.4.2 Patch Effect in the Decay Volume . . . . . . . . . . . . . . . . . . 194
6.4.3 Unexpected Proton Reflections and Influence on a . . . . . . . . . 197
6.4.4 Approaches for the Unexpected Proton Reflections . . . . . . . . . 203
6.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.5.1 Investigations of the Patch Effect and Influence on a . . . . . . . . 205
6.5.2 Improvements for a Further Beam Time . . . . . . . . . . . . . . . 206
6.5.3 Upcoming Measurements with aSPECT . . . . . . . . . . . . . . . 207
7 Summary and Outlook 209
7.1 Statistical and Systematic Limits . . . . . . . . . . . . . . . . . . . . . . . 209
7.2 Improvements for a Further Beam Time . . . . . . . . . . . . . . . . . . . 210
7.3 Upcoming Measurements with aSPECT . . . . . . . . . . . . . . . . . . . 211
7.4 The Future with PERC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A Design of an Anti-Magnetic Screen 213
A.1 The Shielding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2 Axially Symmetric Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.3 Non-axially Symmetric Shielding . . . . . . . . . . . . . . . . . . . . . . . 215
A.4 Electromagnetic Force Calculation . . . . . . . . . . . . . . . . . . . . . . 218
A.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
xii Contents
B Details of the Kelvin Probe Samples 223
C The New Facility PERC 229
C.1 Measurement Principles and Instrument . . . . . . . . . . . . . . . . . . . 229
C.2 Measurement Uncertainties and Systematics . . . . . . . . . . . . . . . . . 230
C.3 Dominant Uncertainties in the Analysis of Protons . . . . . . . . . . . . . 230
Bibliography 233
Erklärung 247
Acknowledgments 249
Curriculum Vitæ 251
List of Figures
1.1 Determination of λ = LA/LV from neutron decay . . . . . . . . . . . . . . 4
2.1 Neutron decay at the quark level . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Illustration of the angular correlation coefficients . . . . . . . . . . . . . . 11
2.3 Theoretical proton recoil spectrum . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Theoretical electron and electron-antineutrino spectra . . . . . . . . . . . 18
2.5 Influence of several corrections on the proton and lepton spectra . . . . . 18
2.6 Determination of |Vud| and λ: Limits from neutron decay . . . . . . . . . 25
2.7 Limits on left-handed scalar and tensor currents from neutron decay . . . 27
2.8 Limits on right-handed scalar and tensor currents from neutron decay . . 29
2.9 Limits on hypothetical W ′ bosons from neutron decay . . . . . . . . . . . 30
2.10 Limits on LH scalar and tensor currents from neutron and nuclear decays 33
2.11 The experiment of Grigor’ev et al. . . . . . . . . . . . . . . . . . . . . . . 36
2.12 The experiment of Stratowa et al. . . . . . . . . . . . . . . . . . . . . . . . 37
2.13 The experiment of Byrne et al. . . . . . . . . . . . . . . . . . . . . . . . . 37
2.14 Sketch of the electromagnetic set-up of aSPECT at the FRM II . . . . . . 38
2.15 Background problems of aSPECT at the FRM II . . . . . . . . . . . . . . 39
2.16 World average of the neutrino-electron correlation coefficient a . . . . . . . 40
2.17 Sketch and Measurement principle of aCORN . . . . . . . . . . . . . . . . 40
2.18 Sketch and Measurement principle of Nab . . . . . . . . . . . . . . . . . . 41
3.1 Scheme of the aSPECT experiment . . . . . . . . . . . . . . . . . . . . . . 44
3.2 The adiabatic transmission energy . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Adiabatic transmission function . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Theoretical integral proton spectrum . . . . . . . . . . . . . . . . . . . . . 51
3.5 Dependence of a on the accuracy in rB and UA . . . . . . . . . . . . . . . 52
3.6 Dependence of a on the accuracy in UA for a set of barrier voltages . . . . 52
3.7 Sketch of the electromagnetic set-up of aSPECT at the ILL . . . . . . . . 54
3.8 Magnetic field and Electrostatic potential along the z-axis . . . . . . . . . 54
3.9 Photographs of the DV and the electrostatic mirror electrodes . . . . . . . 56
3.10 Possible electron and ion traps at the FRM II . . . . . . . . . . . . . . . . 57
3.11 Electrostatic potential in the x-y-plane of the wire system . . . . . . . . . 58
3.12 Photographs of the high voltage electrodes . . . . . . . . . . . . . . . . . . 59
3.13 Electrostatic potential of the upper dipole electrode at the FRM II . . . . 60
3.14 Electrostatic potential of the upper dipole electrode at the ILL . . . . . . 60
3.15 Electric field strength of the detector high voltage electrode . . . . . . . . 61
3.16 Magnetic fields in the DV and the AP . . . . . . . . . . . . . . . . . . . . 63
3.17 Working principle of a silicon drift detector . . . . . . . . . . . . . . . . . 65
xiii
xiv LIST OF FIGURES
3.18 Detector chip - Photograph and Electromagnetic set-up . . . . . . . . . . 66
3.19 Pulse height spectra measured with the silicon PIN diode and the SDD . . 67
3.20 Mechanical set-up of the detection system . . . . . . . . . . . . . . . . . . 68
3.21 Illustration of the trigger algorithm . . . . . . . . . . . . . . . . . . . . . . 69
3.22 Dependence of a on the magnetic field gradient in the DV . . . . . . . . . 71
3.23 Illustration of the edge effect . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.24 Probability of backscattering of decay protons . . . . . . . . . . . . . . . . 80
4.1 Sketch of the experimental set-up at the instrument PF1b of the ILL . . . 82
4.2 Photographs of the experimental set-up at PF1b of the ILL . . . . . . . . 84
4.3 Neutron beam profiles in front of the entrance and behind the exit window 86
4.4 Neutron beam profiles for the 20mm and the 5mm wide aperture . . . . . 87
4.5 Magnetic field profiles along the z-axis, in the DV, and in the AP . . . . . 88
4.6 Energy-calibration spectrum of the SDD with 133Ba . . . . . . . . . . . . 92
5.1 Typical neutron decay events with their resulting fits . . . . . . . . . . . . 96
5.2 Events sorted incorrectly into fit status 1 with their fit and derivative . . . 99
5.3 Comparison of fitted with non-fitted pulse height spectra . . . . . . . . . . 100
5.4 Dependence of a on the electronics dead time . . . . . . . . . . . . . . . . 101
5.5 Dependence of a on a wrongly corrected dead time . . . . . . . . . . . . . 103
5.6 Proton TOF versus proton polar emission angle . . . . . . . . . . . . . . . 104
5.7 Electron versus proton polar emission angle . . . . . . . . . . . . . . . . . 104
5.8 Subtraction of the background from the pulse height spectra . . . . . . . . 105
5.9 Integral proton spectrum and Fit residuals . . . . . . . . . . . . . . . . . . 107
5.10 Dependence of the background without neutron beam on UA . . . . . . . 107
5.11 Dependence of a on uncertainty and slope of the background count rate . 108
5.12 Background spectra without neutron beam and for different types of ions . 109
5.13 Influence of the lower dipole potential on the background count rate . . . 111
5.14 Background spectra for a second analyzing plane . . . . . . . . . . . . . . 111
5.15 Influence of an electric field gradient in the DV on the proton count rates 113
5.16 Dependence of a on the lower integration limit . . . . . . . . . . . . . . . 115
5.17 Pulse height spectra of events after high-energy electrons and TOF spectra 117
5.18 Electron versus proton kinetic energy . . . . . . . . . . . . . . . . . . . . . 117
5.19 Proton event after an electron and Baseline value distribution . . . . . . . 118
5.20 Reduced trigger range and Dependence of the pulse height on the baseline 119
5.21 Illustration of the saturation of the preamplifier . . . . . . . . . . . . . . . 121
5.22 Influence of the preamplifier saturation on the pulse height spectra . . . . 122
5.23 Fraction of electrons that saturate the preamplifier and Altered dead time 123
5.24 Flow chart of the MC simulation to reproduce the saturation effect . . . . 124
5.25 Dependence of a on an artificially increased dead time . . . . . . . . . . . 126
5.26 Time difference spectra with and without electrostatic mirror . . . . . . . 128
5.27 Influence of random coincidences and Gaussian broadening on TOF spectra 129
5.28 Magnetic field and Electron polar angles in the bottom of the spectrometer 130
5.29 Electron trajectories inside an heat shield . . . . . . . . . . . . . . . . . . 132
5.30 Backscatter probability from the bottom of the spectrometer . . . . . . . 133
5.31 Angular distribution of backscattered electrons from copper . . . . . . . . 134
5.32 Energy and angular distributions of backscattered electrons . . . . . . . . 135
5.33 Time difference spectra considering the backscattering of decay electrons . 136
LIST OF FIGURES xv
5.34 Reactor power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.35 Possible background from the electrostatic mirror . . . . . . . . . . . . . . 139
5.36 Sketch of the collimation system inside the spectrometer . . . . . . . . . . 141
5.37 Effect of a charging of the collimation system . . . . . . . . . . . . . . . . 142
5.38 Count rate ratio and Pulse height spectra with and without mirror . . . . 145
5.39 Dependence of the count rate ratio with and without mirror on UA . . . . 146
5.40 Dependence of the proton count rate on the upper dipole potential. . . . . 147
5.41 Dependence of a on the maximum polar angle accepted . . . . . . . . . . 150
5.42 Position of the proton detector . . . . . . . . . . . . . . . . . . . . . . . . 152
5.43 Dependence of count rates with additional aperture on upper dipole drift . 153
5.44 Uncertainty in the position of the detector due to the beam profile . . . . 154
5.45 Uncertainty in the position of the proton detector due to its rotation angle 155
5.46 Dependence of the proton count rates on the lower dipole drift . . . . . . 155
5.47 Dependence of the drift on the proton’s momentum . . . . . . . . . . . . . 157
5.48 Dependence of a on the position of the proton detector . . . . . . . . . . . 159
5.49 Uncertainty in a due to the knowledge of the neutron beam profile . . . . 160
5.50 Dependence of the proton peak on the location on the proton detector . . 160
5.51 Dependence of a on the location on the proton detector . . . . . . . . . . 161
6.1 Cylindrical sample electrode - Photograph and WF topography . . . . . . 168
6.2 Ambient scanning Kelvin Probe system SKP5050 . . . . . . . . . . . . . . 170
6.3 Electron energy level diagrams of two conducting specimens . . . . . . . . 171
6.4 Kelvin Probe signal changes over time . . . . . . . . . . . . . . . . . . . . 172
6.5 Peak-to-peak voltage versus backing potential . . . . . . . . . . . . . . . . 173
6.6 Work function versus surface roughness of copper . . . . . . . . . . . . . . 175
6.7 Electroplated Pt (Cu25) - Photograph, WF topography and histogram . . 179
6.8 Electroplated Pt (Cu25) - High resolution plot and Gradient topography . 180
6.9 Electroplated Pt (Cu25) - Temporal stability over one day . . . . . . . . . 180
6.10 Electroplated Au (Cu12) - Temporal stability over months . . . . . . . . . 182
6.11 Electroplated Au (Cu28) - Photograph and WF topography . . . . . . . . 184
6.12 Electroplated Au (Cu33) - Photograph . . . . . . . . . . . . . . . . . . . . 185
6.13 Surface roughness profiles of copper samples and Perthometer parameters 188
6.14 Surface roughness profiles of titanium samples . . . . . . . . . . . . . . . . 189
6.15 Influence of intermediate layers on corrosion behavior of platinized samples 190
6.16 Sputtered Au (Cu7 and Cu8) - WF topography . . . . . . . . . . . . . . . 191
6.17 Sputtered Au (Cu22) - SEM analysis . . . . . . . . . . . . . . . . . . . . . 191
6.18 Effect of one patch in the analyzing plane . . . . . . . . . . . . . . . . . . 193
6.19 Effect of several patches in the analyzing plane . . . . . . . . . . . . . . . 194
6.20 Effect of one patch in the decay volume . . . . . . . . . . . . . . . . . . . 195
6.21 Effect of various WF inhomogeneities in the decay volume . . . . . . . . . 196
6.22 Influence of WF inhomogeneities on the transmission function . . . . . . . 200
6.23 Dependence of a on WF inhomogeneities . . . . . . . . . . . . . . . . . . . 201
6.24 Small electric field gradient in the DV . . . . . . . . . . . . . . . . . . . . 203
6.25 Dependence of a on an electric mirror above the DV . . . . . . . . . . . . 204
7.1 Scheme of the new facility PERC . . . . . . . . . . . . . . . . . . . . . . . 212
A.1 The magnetic field for a 2D shield made of RTM3 . . . . . . . . . . . . . . 215
xvi LIST OF FIGURES
A.2 The final design for passive shielding . . . . . . . . . . . . . . . . . . . . . 216
A.3 The magnetic field in the symmetry plane that cuts through a pillar . . . 217
A.4 Expected influence on the internal magnetic field . . . . . . . . . . . . . . 218
A.5 Setup of the spectrometer aSPECT inside the anti-magnetic screen . . . . 220
A.6 Influence on the exterior magnetic field . . . . . . . . . . . . . . . . . . . . 221
A.7 Influence on the internal magnetic field . . . . . . . . . . . . . . . . . . . . 221
C.1 Transmission function for aSPECT as detection system for PERC . . . . . 230
List of Tables
2.1 Ft values of superallowed beta decays . . . . . . . . . . . . . . . . . . . . 32
3.1 Typical voltage settings of the electrodes . . . . . . . . . . . . . . . . . . . 55
3.2 Dependence of a on non-adiabatic proton motion . . . . . . . . . . . . . . 73
3.3 Critical pressure values of elastic p-H2 scattering . . . . . . . . . . . . . . 74
3.4 Critical pressure values of charge exchange processes . . . . . . . . . . . . 75
4.1 Range of the magnetic field in the DV and the AP . . . . . . . . . . . . . 89
4.2 Typical voltage settings of the analyzing plane electrode . . . . . . . . . . 91
5.1 Limits of the fit parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Overview of the fit states . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Influence of an electric field gradient in the DV on the proton count rate . 113
5.4 Influence of the mirror potential on the proton count rates . . . . . . . . . 137
5.5 Dependence of the proton count rate on the electrostatic mirror potential 144
5.6 Mean proton drift for different settings of the spectrometer . . . . . . . . 158
5.7 Dependence of a on edge effect for different settings of the spectrometer . 162
5.8 Uncorrected values for a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.9 Corrections and uncertainties on a . . . . . . . . . . . . . . . . . . . . . . 165
6.1 Electron work function of selected elements . . . . . . . . . . . . . . . . . 169
6.2 Summary of Kelvin Probe scans . . . . . . . . . . . . . . . . . . . . . . . . 177
6.3 Temporal stability of Kelvin Probe scans . . . . . . . . . . . . . . . . . . . 181
6.4 Reproducibility of Kelvin Probe samples . . . . . . . . . . . . . . . . . . . 183
6.5 Surface roughness of Kelvin Probe samples . . . . . . . . . . . . . . . . . . 186
6.6 Possible WF inhomogeneities in the DV . . . . . . . . . . . . . . . . . . . 202
A.1 The electromagnetic forces onto an eighth of the coils . . . . . . . . . . . . 219
B.1 Details of the Kelvin Probe samples . . . . . . . . . . . . . . . . . . . . . 224
xvii
xviii LIST OF TABLES
Chapter 1
Introduction
”In 1932, [James] Chadwick made a fundamental discovery in the domain of nuclear science:
he proved the existence of neutrons - elementary particles devoid of any electrical charge
[8]. In contrast with the helium nuclei (alpha rays) which are charged, and therefore
repelled by the considerable electrical forces present in the nuclei of heavy atoms, this
new tool in atomic disintegration need not overcome any electric barrier and is capable
of penetrating and splitting the nuclei of even the heaviest elements. Chadwick in this
way prepared the way towards the fission of uranium 235 and towards the creation of the
atomic bomb. For this epoch-making discovery he was awarded the Hughes Medal of the
Royal Society in 1932, and subsequently the Nobel Prize for Physics in 1935.”, from Nobel
Lectures, Physics 1922-1941 [9].
The present thesis deals with the decay of free neutrons. A neutron (n) is a little more
massive than the proton (p), by ∆ = mn −mp = 1293.333(33) keV/c2, and in free space
it decays weakly into a proton, an electron (e), and an electron-antineutrino (ν̄e):
n −τ→n p + e− + ν̄e +Q, (1.1)
with a mean lifetime of τn = (881.5 ± 1.5) s [10]. Here, the released energy is given by
Q = (mn −mp −me −m )·c2ν = 782.334(33) keV [10]. At the quark level, one of the down
quarks in the neutron decays to an up quark, emitting a virtual W gauge boson, which
further decays into an electron and an electron-antineutrino (see Fig. 2.1). As the simplest
nuclear beta decay, the free neutron provides an excellent framework for the study of the
structure and nature of the weak interaction.
The Standard Model (SM) of elementary particles and fields is the mathematical the-
ory that describes the weak, electromagnetic, and strong interactions between leptons and
quarks: In 1960, Sheldon Glashow developed a theory which unifies the weak and the
electromagnetic interactions [11]. In 1967, Steven Weinberg and Abdus Salam, incorpo-
rated the Higgs mechanism1 into Glashow’s electroweak theory [13, 14] (see also [15]). The
Glashow-Salam-Weinberg (GSW) theory, along with QCD2, constitute the SM. In 1973,
the weak neutral currents have been discovered in the Gargamelle3 bubble chamber at
CERN [16]. Then, the GSW theory became widely accepted, and Glashow, Salam, and
Weinberg shared the 1979 Nobel Prize in Physics for their contributions to the theory.
1The Higgs mechanism refers to the generation of masses for theW± and Z weak gauge bosons through
electroweak symmetry breaking [12]. Currently, experiments at the Large Hadron Collider at CERN are
searching for Higgs bosons.
2Quantum Chromodynamics (QCD) is a theory of the strong interaction.
3Gargamelle was a bubble chamber principally designed for the detection at CERN of neutrinos.
1
2 CHAPTER 1. INTRODUCTION
The SM has successfully explained a large number of experimental results and precisely
predicted a wide variety of phenomena. So far, it is the most successful theory of particle
physics. However, it is not a complete theory of all four fundamental forces, because it
does not incorporate general relativity. Physics beyond the SM attempts to explain the
deficiencies of the SM, such as the origin of mass, the strong CP problem, the hierarchy
problem, neutrino oscillations (and their non-zero masses), matter-antimatter asymmetry,
and the origins of dark matter and dark energy. Theories beyond the SM include various
extensions of the SM through supersymmetry (SUSY), such as the Minimal Supersym-
metric Standard Model (MSSM), or entirely novel explanations, such as string theory and
extra dimensions.
High-precision measurements of observables in neutron beta decay address a number
of questions which are at the forefront of particle physics [17–19], and are generally com-
plementary to direct searches in high-energy physics. Main emphasis lies on the search
for new physics beyond the SM. Possible extensions require new symmetry concepts like
left-right symmetry, fundamental fermion compositeness, new particles, leptoquarks, su-
persymmetry, supergravity, or many more [20, 21]. Free neutron decay is therefore a very
active field, with a number of new measurements underway worldwide. For recent reviews
see Refs. [18, 19, 22, 23]. With high-precision measurements of angular correlations several
symmetry tests based on neutron beta decay data become competitive, cf. Sec. 2.3 and
Ref. [6].
In the modern form of the SM, the differential decay rate of neutrons can be written
as [24]:
2 2
d3
1 G
Γ = F
|Vud|
p E (E − E )2 dE dΩ dΩ ×
(2[π)5 2 e e 0 e ( e e ν )]
p · p m 〈s 〉 p p
ξ 1 + e ν ea + b + n eA + νB + . . . . (1.2)
EeEν Ee sn Ee Eν
Here, GF is the Fermi weak coupling constant, Vud is the upper left element of the Cabibbo-
Kobayashi-Maskawa (CKM) quark-mixing matrix [25, 26], pe, pν , Ee, and Eν are the
electron (neutrino) momenta and total energies, respectively, E0 is the electron spectrum
endpoint total energy, me is the electron mass, sn is the neutron spin, and the Ωi denote
solid angles. Quantity ξ is a factor inversely proportional to the neutron decay rate, a, A,
and B are the angular correlation coefficients, while b is the Fierz interference term. The
neutrino-electron correlation coefficient a and the Fierz term b are measurable in decays
of unpolarized neutrons, while the beta and neutrino asymmetry parameters A and B
require polarized neutrons. A non-vanishing Fierz term b would indicate the existence of
left-handed scalar and tensor interactions.
Another observable is C, the proton asymmetry relative to the neutron spin [27, 28]:
C = −xC(A+B), (1.3)
where xC = 0.27484 is a kinematical factor4.
The present status on the angular correlation coefficients a, A, B, and C, the Fierz
term b, and the neutron lifetime τn is summarized in Ref. [19]. According to the Particle
Data Group’s (PDG) 2011 review [10], the relative errors on a, A, B, and C are 4%,
0.9%, 0.3%, and 1.1%, respectively. Recently, three beta asymmetry experiments have
4Note that we define the proton asymmetry C with the opposite sign compared to [27]. This retains
the convention that a positive asymmetry indicates more particles to be emitted in the spin direction.
3
completed their analyses, namely UCNA [29], PERKEO II [30]5, and PERKEO III [32].
The PERKEO III collaboration improved the uncertainty on A by about a factor of 5
compared to the PDG 2011 average [32] (preliminary). For the non-SM parameter b only
an upper limit has been derived.
Within the framework of the SM, neutron beta decay is described as a purely left-
handed, V−A interaction. Then, b = 0 and the correlation coefficients a, A, B, and C
depend only on the ratio λ = LA/LV of the weak axial-vector (LAGFVud) to the vector
(LVGFVud) coupling constant:
1− |λ|2 − |λ|
2 + λ |λ|2 − λ 4λ
a = 2 , A = 2 2 , B = 2 2 , and C = xC 2 . (1.4)1 + 3 |λ| 1 + 3 |λ| 1 + 3 |λ| 1 + 3 |λ|
Near the value λ = −1.27, the sensitivities of a, A, B, and C to λ are6:
da dA dB dC
= 0.298, = 0.374, = 0.076, and = −0.124. (1.5)
dλ dλ dλ dλ
The size of the weak coupling constants is important for applications in cosmology (e.g.,
primordial nucleosynthesis), astronomy (e.g., solar physics), and particle physics (e.g.,
neutrino detectors, neutrino scattering) [18, 19, 22]. The value of λ can be determined
from several independent neutron decay observables, each with different sensitivity to non-
SM physics. Comparing the various values of λ therefore provides an important test of
the validity of the SM.
The neutron lifetime τn is inversely proportional to |V 2ud| (1 + 3|λ|2) [35]. Hence, inde-
pendent measurements of τn and of an observable sensitive to λ allow the determination
of |Vud|. Along with |Vus| and |Vub| from K-meson and B-meson decays, respectively, the
unitarity of the CKM matrix is tested [10, 36], which in turn is a test of self-consistency of
the SM. Presently, |Vud| cannot be determined reliably from neutron decay data because
of very inconsistent experimental results for the neutron lifetime [10] (see also Sec. 2.2.1).
At present, the values of λ with the smallest quoted errors come from measurements
of the beta asymmetry parameter A. Unfortunately, the measurements of A with the
lowest quoted errors are in disagreement with earlier experiments, as can be seen from
Fig. 1.1. The origin of this discrepancy is unknown, although large corrections (in the
15 − 30% range) had to be made for, e.g., neutron polarization [37–39]. One of the
most serious problems when working with polarized neutrons is the accurate knowledge
of the average neutron beam polarization, where presently the error is on the 10−3 level
[44]. A different approach to determine λ, which does not require polarized neutrons, is to
measure the neutrino-electron correlation coefficient a. A measurement of a is independent
of possible unknown errors in A and has entirely different systematics. However, previous
measurements of a have been limited by systematic uncertainties to 5% [40, 41], and
since the late 1970s there have been no essential improvements. In addition, a precise
comparison of a and A can set strict limits on possible conserved-vector-current violation
and second class currents in neutron beta decay [45]. Hence, there is a great interest in
improving the uncertainty in a to less than 1%.
The neutron decay spectrometer aSPECT [46] has been built to perform precise mea-
surements of the correlation coefficients a [47] and C [48], by measuring the proton recoil
5Publication of the result of the last PERKEO II run [31] is underway.
6Please note that in Refs. [33, 34] the sensitivities of a, A, and B are misstated with negative sign.
4 CHAPTER 1. INTRODUCTION
Figure 1.1: Ratio λ = LA/LV of the weak axial-vector to the vector coupling constant, derived
from measurements of correlations a (red circles, [40, 41]) and A (green squares,
[18, 29, 32, 37–39, 42, 43]). In the case of PERKEO III, the error bar includes the total
uncertainty of the measurement (blue square) together with the still blinded correc-
tions (polarization, magnetic mirror effect). The black circle rightmost shows how a
0.3% measurement of a would contribute to the determination of λ. For comparison,
the grey bar represents the Particle Data Group’s (PDG) 2011 average [10], where
only the measurements of the correlation A are used, except for the latest results of
PERKEO II and III.
spectrum in the decay of unpolarized or polarized neutrons, respectively. Since it is hard
to detect the neutrino, we infer a from the shape of the proton recoil spectrum. We aim to
improve the uncertainty in a and C to 0.3% [47] and 0.1% [48], respectively. Figure 1.1
shows how a 0.3% measurement of a would contribute to the determination of λ.
In our first beam time at the Forschungsneutronenquelle Heinz Maier-Leibnitz in Mu-
nich, Germany, background instabilities due to particle trapping and the electronic noise
level of the proton detector had prevented us from presenting a new value for a from
this beam time [33, 49]. In the latest beam time at the Institut Laue-Langevin (ILL) in
Grenoble, France, the trapped particle background has been reduced sufficiently and the
electronic noise problem has essentially been solved [5, 50]. For the first time, a silicon drift
detector was used [51]. As a result of the data analysis, we identified and fixed a problem
in the detector electronics which caused a significant systematic error. The present thesis
focused on the theoretical and experimental investigation of several different systematic
effects. For details on the raw data analysis, the reader is referred to the theses of M.
Simson [52] and M. Borg [34]. Details on detector and electronics tests and measurements
of the neutron beam profile can be found in Refs. [34, 52], respectively. Measurements of
the magnetic field and the development of an online nuclear magnetic resonance system
are presented in the thesis of F. Ayala Guardia [53].
In Chap. 2, the theoretical basics, searches for physics beyond the SM, and previous
and competing projects are presented. The measurement principles and systematics and
the experimental set-up of aSPECT are described in Chap. 3. In Chaps. 4 and 5, the
measurements at the ILL and the data analysis are discussed. Investigations of the patch
effect, in order to improve the aSPECT spectrometer, are presented in Chap. 6. In Chap. 7,
the results of the present thesis are discussed. The Apps. A to C contain additional
information about the Kelvin probe samples, the design of a magnetic field return, and
the new facility PERC.
Chapter 2
Neutron Beta Decay
Our understanding of the neutron has improved considerably through its description
within the framework of the Standard Model (SM) of elementary particles and fields.
The neutron consists of three quarks: two down quarks (d) and an up quark (u). It is
stable under the strong and electromagnetic interactions, but unstable under the weak
interaction: Through the emission of a virtual W gauge boson, a down quark can decay
into an up quark, as shown in Fig. 2.1.
Neutron decay experiments provide one of the most sensitive means for determining the
weak vector (LVGFVud) and axial-vector (LAGFVud) coupling constants, and the element
Vud of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix. Here GF is the
Fermi weak coupling constant. The value of LV is important for testing the conserved
vector current (CVC) hypothesis. The size of the weak coupling constants is important
for applications in cosmology, astronomy, and particle physics [18, 19, 22].
In the framework of the SM, the CVC hypothesis requires LV = 1 for zero momentum
transfer. Therefore, neutron beta decay is described by two parameters only, λ = LA/LV
and Vud. The neutron lifetime τn is inversely proportional to Eq. (2.36) |V 2ud| (1 + 3|λ|2).
Hence, independent measurements of τn and of an observable sensitive to λ, allow the
determination of Vud. The value of λ can be determined from several independent neutron
decay observables, introduced in Sec. 2.1.4. Each observable brings a different sensitivity
to non-SM physics, such that comparing the various values of λ provides an important
test of the validity of the SM. Of particular interest in this context is the search for scalar
and tensor interactions, discussed in Secs. 2.1.6 and 2.3. These interactions can be caused,
e.g., by leptoquarks or charged Higgs bosons [20]. There are a number of other extensions
of the SM. In Secs. 2.1.6 and 2.3.1 we discuss a particular kind of V+A interactions,
the manifest left-right symmetric (MLRS) models, with the SU(2)L×SU(2)R×U(1)B−L
gauge group and right-handed charged current, approximately realized with a minimal
Higgs sector.
After a short theoretical introduction to neutron beta decay, tests of and searches for
physics beyond the SM in neutron decay are presented in this chapter. These searches
have already been presented in Ref. [6]. I will closely follow the description therein and
update the current status. The main emphasis of aSPECT and in particular of this thesis
lies on the determination of the neutrino-electron angular correlation coefficient a. Hence,
we also introduce the measurable parameters and spectra of neutron beta decay. Finally,
we will discuss previous measurements of a and upcoming neutron decay experiments.
For more details on neutron physics and the theory of electroweak interaction, the
reader is referred to recent reviews [18, 19, 54] and standard textbooks [17, 55–57].
5
6 CHAPTER 2. NEUTRON BETA DECAY
Figure 2.1: Neutron decay at the quark level: One of the down quarks (d, green) in the neutron
(n) decays to an up quark (u, red), through the emission of a virtual W gauge boson,
which further decays into an electron (e) and an electron-antineutrino (ν̄e). The gluons
(Gauge bosons that participate in the strong interaction between quarks in QCD,
white) hold the quarks together to form protons (p) and neutrons.
2.1 Theory of Neutron Beta Decay
Neutron beta decay is described by the weak interaction. The weak interaction, in turn,
is described by the SM of elementary particles and fields.
2.1.1 The Standard Model
The SM claims that the matter in the Universe consists of elementary fermions, interacting
through fields they cause. The particles related with the interaction fields are the gauge
bosons. Regarding the interaction fields, the SM excludes from consideration general
relativity.
The spin-1 gauge bosons of the weak interaction fields between fermions are the charged
W± and the neutral Z bosons. The gauge boson of the electromagnetic interaction field
between electrically charged fermions are the massless photons (γ’s). And the postulated
8 gauge bosons of the strong interaction field are the gluons.
Two types of elementary spin-12 fermions are distinguished: 6 leptons and 6 quarks.
Leptons interact only through the electromagnetic interaction, provided that they are
charged, and the weak interaction. Quarks interact through all three fundamental forces
included in the SM.
The Higgs mechanism refers to the generation of masses for theW± and Z weak gauge
bosons through electroweak symmetry breaking [12]. The existence of the spin-0 Higgs
boson is postulated to resolve inconsistencies in theoretical physics. So far, the Higgs boson
is the only SM particle which has not been observed experimentally. Present experiments
suggest a Higgs mass of (115− 200)GeV/c2 [10] (more exact, the D0
 collaboration [58]).
Although the SM is theoretically self-consistent, it has several deficiencies like the
strong CP problem. In 1964, James Cronin and Val Fitch discovered CP violation in the
2.1. THEORY OF NEUTRON BETA DECAY 7
decays of neutral K-mesons. This discovery earned both the 1980 Nobel Prize in Physics,
as it showed that weak interaction violates not only the charge-conjugation symmetry C
between particles and antiparticles and the parity P, but also their combination. Parity
violation was suggested in 1956 by Tsung-Dao Lee and Chen Ning Yang [59] and shortly
after demonstrated by Chien-Shiung Wu [60] in the beta decay of 60Co, earning Yang
and Lee the 1957 Nobel Prize in Physics. The CP violation is incorporated in the SM
by including a complex phase in the CKM matrix, describing quark mixing (see also the
following section). In such theory, a necessary condition for the appearance of the complex
phase is the presence of at least three quarks generations. The strong CP problem, in
turn, is the question why quantum chromodynamic (QCD) does not seem to break the
CP symmetry.
There are a number of attempts to explain the deficiencies of the SM; a brief overview
is given in Chap. 1. In Secs. 2.1.6 and 2.3, we discuss extensions of the SM addressing
these deficiencies. We note that the strong CP problem may also be solved within a theory
of quantum gravity.
For an historical overview of the theory of the SM, the reader is referred to standard
textbooks; a brief historical overview is found in Chap. 1.
2.1.2 Weak Interaction in the SM
For daily life, weak interactions are most noticeable in nuclear beta decay and in the p-p
chain reaction1 that dominates the energy generation in the sun. Beta decay also makes
radiocarbon dating possible, as 14C decays through the weak interaction to 14N.
A neutron, e.g., is a little more massive than a proton, but it cannot decay into a
proton without changing the flavor2 of one of its two down quarks into up. Neither the
electromagnetic nor the strong interaction permit flavor changing, so this must be carried
out by the weak force. In fact, a down quark in the neutron can change into an up
quark through the emission of a virtual W gauge boson, which, in turn, decays into an
electron and an electron-antineutrino. Figure 2.1 shows an (illustrated) Feynman diagram
of neutron beta decay.
The weak force was first described, in the 1930s, by Enrico Fermi’s theory of a contact
four-fermion (vector) interaction [61], in which the hadronic and leptonic currents interact
at one vertex. In 1936, his theory was extended by George Gamow and Edward Teller
to describe transitions which required the introduction of other possible (scalar) Lorentz
invariants3 [62]. In 1958, George Sudarshan and Robert Marshak [63], and also inde-
pendently Richard Feynman and Murray Gell-Mann [64], incorporated the assumption
of maximal parity violation into the theory, and determined the correct tensor structure
(vector minus axial vector, V−A) of the four-fermion interaction. The Hamiltonian of the
V−A theory has the form of a current-current interaction4
H GV−A = √FJ†µ · Jµ + h.c., (2.1)2
1The proton-proton chain reaction is one of several fusion reactions by which stars convert 1H to (2D
and then) 4He. The primary alternative is the CNO cycle.
2There are six types, also known as flavors, of quarks: up, down, charm, strange, top, and bottom.
3Quantities that do not change due to a Lorentz transformation, i.e., quantities that are independent
of the inertial frame.
4In Eq.(2.1) the h.c. stands for the hermitian conjugate.
8 CHAPTER 2. NEUTRON BETA DECAY
where GF is the Fermi weak coupling constant. The current Jµ contains a hadronic and a
leptonic contribution
J = Jhad + J lepµ µ µ . (2.2)
The strength of Fermi’s interaction is given by the Fermi weak coupling constant GF,
evaluated from the muon lifetime [10]. In modern terms [57]:
G g2 2F = √ e= √ = 1.16637(1)× 10−5GeV−2, (2.3)
(~c)3 4 2m2 2W 4 2mW sin
2 θW
where g is the coupling constant of the weak interaction, e is the electron charge, mW is
the mass of the W gauge boson, and θW is the Weinberg angle5. Due to the large mass of
the gauge boson, weak decay is much more unlikely than electromagnetic or strong decay,
and hence occurs less rapidly. For example, a free neutron lives about 15minutes (see also
Sec. 2.2.1).
The relative strength of the weak interaction in pure leptonic, in semileptonic, and in
pure hadronic processes are not identical. This has been incorporated into the theory with
the mechanism of quark mixing: In 1963, Nicola Cabibbo introduced the Cabibbo angle,
θC, [25] to preserve the universality of the weak interaction. In his theory, the weak eigen-
states of the quarks are postulated to differ from the eigenstates of the electromagnetic
and strong interaction, which define the mass eigenstates. In 1973, Makoto Kobayashi and
Toshihide Maskawa showed that CP violation in the weak interaction required more than
two generations of particles [26], effectively predicting the existence of a then unknown
third generation. In 2008, Kobayashi and Maskawa shared one half of the Nobel Prize in
Physics for their work. In the case of the three quark families, the mixing is expressed by
the Cabibbo-Kobayashi-Maskawa (CKM) matrix [25, 26]:
 
 
|d′〉
|s′〉   Vud Vus Vub=  |d〉 Vcd Vcs Vcb |s〉  . (2.4)
|b′〉 Vtd Vts Vtb |b〉
Here, the primes ′ denote the weak eigenstates. For nuclear beta decay, we have
d′ ≈ Vudd = cos θCd. (2.5)
Hence, the weak interaction of the down quark introduces the matrix element Vud into the
amplitude of the hadronic current. We note that the normalization of states requires the
CKM matrix to be unitary. The present status on the matrix elements and the unitarity
of the CKM matrix is summarized in the Particle Data Group’s (PDG) 2011 review [10]
(see also Sec. 2.2.3).
Selection Rules
Beta transitions are classified in allowed and forbidden6 decays. Allowed decays corre-
spond to transitions in which the electron and neutrino do not carry any orbital angular
5The Weinberg angle or weak mixing angle is a parameter in the Weinberg-Salam theory. It is the angle
by which spontaneous symmetry breaking rotates the original W 0 and B0 vector boson plane, generating
the Z0 boson, and the photon (γ).
6Forbidden does not mean that they do not occur in nature.
2.1. THEORY OF NEUTRON BETA DECAY 9
momentum. Their selection rules are:
∆J = Ji − Jf = 0,±1, no 0→ 0 (2.6)
πiπf = +1, (2.7)
where Ji, Jf , πi, and πf are the spin and parity of the initial (i) and final (f) state of the
nucleus, respectively. The parity does not change in such a transition.
The allowed transitions are subdivided into Fermi and Gamow-Teller decays. A tran-
sition mediated by the vector component does not change the spin of the nucleus (the spin
of electron and antineutrino couple to a total spin S = 0) and is called Fermi decay. For
a transition mediated by the axial-vector component the spins of electron and neutrino
couple to S = 1; such a transition is called Gamow-Teller decay.
In the case of Jπii = J
πf
f , the decaying nucleon ends up in the same state where it
originated, e.g., in the decay of 14O to 14N. Decays of this type are called superallowed
decays (see also Secs. 2.2 and 2.3).
2.1.3 Neutron Beta Decay in the SM
Fermi’s golden rule states that the transition rate Γ, i.e., the inverse of the lifetime, from
an initial to a final state is given by
2π
Γ = |M |2fi ρ, (2.8)~
whereMfi is the matrix element describing the interaction and ρ is the phase-space den-
sity of the final states. The matrix element Mfi describing neutron beta decay can be
constructed as a four-fermion interaction composed of hadronic and leptonic matrix ele-
ments. Assuming that vector (V), axial-vector (A), scalar (S), and tensor (T) currents
are involved, the decay matrix element can be written as a sum of left-handed7 (LH) and
right-handed (RH)∑matrix elements:
8
M 2G√FVud 〈 | | 〉〈 −| 1− γ5 | 〉 〈 | | 〉〈 −| 1 + γ5fi = Lj p Γj n e Γj νe +Rj p Γj n e Γj |ν 〉, (2.9)
2 2 2
e
j∈{V,A,S,T}
where the four types of currents are defined by the operators:
i[γ , γ ]
ΓV = γµ, ΓA = iγµγ5, ΓS = 1, and Γ =
µ√ νT . (2.10)
2 2
Here, the γµ are the Dirac (gamma) matrices. The coupling constants to left-handed
and right-handed neutrinos are denoted by Lj and Rj , respectively. This parametrization
was introduced in Ref. [27] in order to highlight the handedness of the neutrino in the
participating V,A,S,T currents. The Lj and Rj coupling constants are linear combinations
of the coupling constants, Cj and C ′j , which were defined in earlier work [59]:
G V
= √F ud ′ G( + ) = √FVudCj Lj Rj , Cj (Lj −Rj), for j = V, A, S, T. (2.11)2 2
7The helicity h = ~s · ~p of a particle is right-handed if the direction of its spin ~s is the same as the
s p
direction of its motion ~p (h = +1). It is left-handed if the directions of spin and motion are opposite
(h = −1).
8In Eq. (2.9) 1−γ5 and 1+γ5 are the operators that project the neutrino field to its left-handed and
2 2
right-handed part, respectively.
10 CHAPTER 2. NEUTRON BETA DECAY
We emphasize that Eq. (2.9) is not the most general Hamiltonian that one can con-
struct. In addition to V,A,S,T currents, Lee and Yang [59] allowed the possibility of
pseudo-scalar (P) currents, with operator ΓP = γ5. This yields ten possible couplings
under the assumption that all are real (up to an overall common phase) and 20 couplings
if each is allowed an imaginary component. In the latter case, time-reversal invariance,
T , may be violated. In the non-relativistic limit, the pseudo-scalar hadronic current,
〈p|ΓP|n〉, vanishes [62], therefore we neglect the pseudo-scalar term in our calculations. In
addition, we neglect effects of T violation, i.e., we consider the remaining 8 couplings to
be real.
In our tests of the SM and searches for physics beyond the SM (cf. Secs. 2.2 and
2.3), we are mainly interested in coupling constant ratios. We choose GFVud as a free
parameter, therefore we are allowed to define dLV =
ef 1 (CVC). In the SM, the only non-
vanishing coupling constants are LV = 1 and LA = λ. In more general models, other
coupling constants appear.
2.1.4 Measurable Parameters in Neutron Beta Decay
In neutron decay experiments the outgoing spins are usually not observed. Summing
over these spin quantities, and neglecting the neutrino masses, one can evaluate the triple
differential decay rate to be [24]9:
3 1 G
2
F|V 2d Γ = [ ud
|
peEe (E0 − E 2e) d(EedΩedΩν )] (2.12)(2π)5 2
× pe · p1 + ν me 〈sn〉 p+ + e p+ ν pe × pνξ a b A B +D ,
EeEν Ee sn Ee Eν EeEν
where10 [27]:
ξ = |L |2V + 3|L 2 2 2 2 2 2 2A| + |LS| + 3|LT| + |RV| + 3|RA| + |RS| + 3|RT| , (2.13)
ξa = |L |2 − |L |2 − |L |2 + |L |2V A S T + |R |2V − |R |2 − |R |2A S + |R |2T , (2.14)
ξb = 2<(L ∗SLV + 3LAL∗ ∗ ∗T +RSRV + 3RART), (2.15)
ξA = 2<(−|L |2A − L ∗VLA + |LT|2 + LSL∗T + |R |2A +RVR∗ − |R |2A T −R ∗SRT),
(2.16)
m
B = B0 +
e
bν , with (2.17)
Ee
ξB0 = 2<(|LA|2 − L ∗VLA + |LT|2 − LSL∗T − |R |2A +RVR∗ − |R |2A T +R ∗SRT),
(2.18)
ξb = 2<(−L L∗ν S A − L L∗ ∗V T + 2LALT +R R∗ +R R∗S A V T − 2RAR∗T), and (2.19)
ξD = 2=(L L∗ − L L∗S T V A +R R∗ ∗S T −RVRA). (2.20)
Here pe, pν , Ee, and Eν are the electron (neutrino) momenta and total energies, respec-
tively, E0 is the maximum electron total energy, me the electron mass, sn the neutron spin,
and the Ωi denote solid angles. Quantities a, A, B, and D are the angular correlation
coefficients, while b is the Fierz interference term. The latter, and the neutrino-electron
correlation coefficient a, are measurable in decays of unpolarized neutrons, while the A and
9In the modern form of the Standard Model.
10Please note that, here, we neglect Coulomb corrections of order α compared to Ref.[65].
2.1. THEORY OF NEUTRON BETA DECAY 11
Figure 2.2: Illustration of the angular correlation coefficients a, A, B, and C. The neutron spin
(green) divides the space into two hemispheres. In this example, proton and electron
are emitted into the same hemisphere. The neutrino is restricted to the opposite
hemisphere due to momentum conservation. The parameter a relates the electron to
the neutrino momentum.
B, the beta and neutrino asymmetry parameters, respectively, require polarized neutrons
(see Fig. 2.2). bν is another Fierz-like parameter, similar to b. We note that a, A, B0, and
D are sensitive to non-SM couplings only in second order, while b and bν depend in first
order on LS and LT. A non-zero Fierz term b would indicate the existence of LH S and T
interactions. A non-vanishing triple correlation coefficient D would violate T invariance.
The most sensitive measurement of D = (−0.96±1.89±1.01)×10−4 in nuclear beta decay
has been conducted in the beta decay of polarized neutrons [66] (see also [67]).
If the electron spin se is observed more correlation coefficients like N and R appear in
the differential decay rate of neutrons [24]:
2 1 G
2 2
d Γ = [ F
|Vud|
peEe (E0 − E 2e)(dEedΩe )] (2.21)(2π)4 2
× m 〈s1 + e + n〉 pe 〈s 〉 〈s 〉 pξ b A + s n n ee N +R × + . . . ,
Ee sn Ee sn sn Ee
where [27, 65]:
ξN = 2<(L ∗ ∗SLA + LV LT + 2LTL∗A +R R∗S A +RVR∗T + 2R R∗T A) (2.22)
m
+ e · 2<(|L 2A| + L ∗V LA + |LT |2 + LSL∗T + |R 2 ∗ 2 ∗A| +RVRA + |RT | +RSRT ),Ee
ξR = 2=(LSL∗A − LV L∗T + 2L ∗ ∗TLA −RSRA +RVR∗T − 2RTR∗A) (2.23)
m
+ eα · 2<(|L 2A| + L ∗V LA − |LT |2 − L ∗ 2SLT − |RA| −R R∗V A + |R |2T +R ∗SRT ).pe
Here, the neutrino momentum was averaged over. We note that N and R depend linearly
on S and T couplings. First experimental values for N = 0.065 ± 0.012 ± 0.004 and
R = 0.006± 0.012± 0.005 have been presented in [68, 69].
Another observable is C, the proton asymmetry relative to the neutron spin. Observ-
ables related to the proton do not appear in Eq. (2.13). However, the proton is kinemat-
ically coupled to the other decay products. Neglecting recoil-order effects and radiative
12 CHAPTER 2. NEUTRON BETA DECAY
corrections, the proton asymmetry parameter C is expressed by [27]:
C = −x (A+B )− x′C 0 Cbν , (2.24)
where xC = 0.27484 and x′C = 0.1978 are kinematical factors
11.
The Angular Correlation Coefficients in the Standard Model
Within the framework of the SM, neutron beta decay is described as a purely left-handed,
V−A interaction. Then, the Fierz terms b = 0 and bν = 0. If we permit the possibility
of T violation, the correlation coefficients a, A, B, C, D, N , and R depend only on the
ratio λ = LA/LV of the weak axial-vector to the vector coupling constant and the phase
angle φ between them:
1− |λ|2
a = 2 , (2.25)1 + 3 |λ|
− |λ|
2 + |λ| cosφ
A = 2 , (2.26)
1 + 3|λ|2
|λ|2 − |λ| cosφ
B = 2 , (2.27)
1 + 3|λ|2
|λ| cosφ
C = 4xC , (2.28)1 + 3|λ|2
|λ| sinφ
D = 2 , (2.29)
1 + 3|λ|2
m |λ|2e · + |λ| cosφ mN = 2 = − e ·A, and (2.30)
Ee 1 + 3|λ|2 Ee
m 2e · |λ| + |λ| cosφ − mR = α 2 = eα ·A, (2.31)
p 1 + 3|λ|2e pe
where α is the fine structure constant. If however T invariance is assumed, =λ = 0 and
the triple correlation coefficient D = 0, and the expressions Eqs. (2.26) to (2.28) for A,
B, and C reduce to the familiar expressions Eq. (1.4). When Coulomb corrections are
neglected, the triple correlation coefficient R = 0.
The Neutron Lifetime
The neutron lifetime τn can be derived from the ratio r of the Ft0
+→0+
Ft values in
superallowed 0+ → 0+ nuclear beta decays to the equivalent quantity in neutron decay,
Ftn:
F 0+→0+t Ft0+→0+ F 0+t →0+
rFt = = = · τ−1, (2.32)Ftn fnt(1 + δ′ ) f nR R ln (2)
where ∫E0
1
fn = dE F (1, E ) p E (E − E )2 [1 +R (E
5 e e e e 0 e 0 e
)] = 1.6887 (2.33)
me
me
11Please note that we define C with the opposite sign compared to Refs. [27, 28] to adhere to the
convention that a positive asymmetry indicates that more particles are emitted in the direction of spin.
2.1. THEORY OF NEUTRON BETA DECAY 13
is a statistical phase-space factor [36]. The factor fn includes a correction for the Coulomb
attraction of the final states known as the Fermi function Eq. (2.52) F (Z,E )12e as well as
smaller recoil corrections Eq. (2.56) R0(Ee). The nucleus-dependent (outer) radiative cor-
rection δ′R, andO(α2) corrections [35, 70, 71], change fn by ∼ 1.5% to fR = 1.71385(34)13.
The corrections implicitly assume the validity of the V−A theory [73]. The dependence
of rFt on the coupling constants Lj and Rj is given in Ref. [27]:
ξ
rFt = , (2.34)
ξF
where14
ξF = 2(|L 2V| + |LS|2 + |RV|2 + |R 2S| ). (2.35)
Within the framework of the SM, the matrix element |Vud| can be determined from
the neutron decay ra∫te Γ respectively the neutron lifetime τn and λ [35, 36]:5 4
τ−1
m c
n = Γ = (d3Γ = )e (|L |2V + 3|L 2 n ′A| )f (1 + δR)(1 + ∆V)2π3~7 R
|V 2ud| 1 + 3|λ|2= , (2.36)
(4908.7± 1.9)s
where ∆VR = 2.361(38)% considers the inner (nucleus-independent) radiative correction
[76].
The present status on the correlation coefficients a, A, B, C, D, N , and R, the Fierz
terms b and bν , and the neutron lifetime τn is summarized in Ref. [19] (see also Sec. 2.2.1).
For D and R and the non-SM parameters b and bν only upper limits have been measured.
In the following, we assume T invariance and therefore D = 0. In addition, we
disregard the recent results for N and R [68, 69], as they lack the precision to have an
impact on our analysis.
2.1.5 The Proton and Lepton Spectra
The main emphasis of aSPECT and in particular of this thesis is on the determination
of the angular correlation coefficient a. For unpolarized neutron decay, we rewrite the
differential decay rate Eq. (2.13):
d3Γ ∝ m1 + aβ cos θeν + eb , (2.37)
Ee
where β = vec is the electron’s velocity in units of the velocity of light and θeν is the angle
between the directions of the electron and the electron-antineutrino. As discussed earlier
12For neutron, and generally nuclear, decays Z denotes the atomic number of the daughter nucleus, i.e.,
Z = 1 for neutron beta decay.
13The most recently published value of fR = 1.71335(15) [72] used fn = 1.6886, and did not include
the corrections by Marciano and Sirlin [35]. Applying the Towner and Hardy prescription for splitting the
radiative corrections [71] increases the uncertainty in fR slightly, to reproduce Eq. (18) in Ref. [35].
14Compared with Ref. [74], here, the factor 2 in Eq. (2.35) considers the nuclear matrix element MF
(see also [75]). For superallowed Fermi beta decay transitions between spin Jπ = 0+√ , isospin T = 1,
states, the Fermi matrix element becomes |M |2 = |M0 |2 V 0 VF F (1 − δC), with MF = 2. Here, δC is the
isospin-symmetry-breaking correction.
14 CHAPTER 2. NEUTRON BETA DECAY
in Chap. 1, the neutrino is hard to detect. Thus, a must be inferred from the electron and
proton momenta instead. One approach is to measure the angular correlation between both
momenta and to derive the correlation coefficient a from it via momentum conservation.
An alternative approach is to infer a from the shape of the proton recoil spectrum.
The Proton Recoil Spectrum
Nachtmann calculated relativistic corrections to the recoil spectrum in neutron beta decay
[77]15. Based on his description of the proton recoil spectrum, Dawber et al. showed that
the spectrum is linearly dependent on the correlation coefficient a [79]:
wp,[Nac68](Tp) ∝ g1(Tp) + a · g2(Tp), (2.38)
where Tp = Ep −mp is the proton kinetic energy. Neglecting Coulomb16 and radiative
corrections,(the functio)ns g1 and g2 a[re (expressed b)y [79]:2√ ]x2 x2− − 4 σ(T )− x2g1(T ) = (1 ) 1 σ(T ) [4(1 + − ) (1− σ(T )) , (2.39)σ(T ) √ σ(T ) 3 σ(T )2 ]x2 x2 4 σ(T )− x2
g2(T ) = 1− 1− σ(T ) 4 1 + − 2σ(T ) − (1− σ(T )) ,
σ(T ) σ(T ) 3 σ(T )
(2.40)
where
σ(T ) = 1− 2Tmn/∆2, (2.41)
[10]
x = me/∆, and ∆ = mn −mp = 1293.333(33)keV/c 2. (2.42)
Here, mn and mp are the masses of the neutron and proton, respectively.
Figure 2.3 shows the influence of the correlation coefficient a on the proton recoil
spectrum. With the recommended value for a = −0.103 [10], this is about a 10% effect.
A positive neutrino-electron correlation a increases the average proton momentum and
shifts the spectral shape to favor higher energy. A negative correlation a has the opposite
effect. This means that we simply measure the shape of wp,[Nac68](T ) and fit it to determine
a. This method has the advantage of requiring the detection of only one particle per decay,
that comes along with a larger event rate. Neutron beam densities are typically low and
the neutron lifetime is long (≈ 15min [10]), so a larger event rate is important for high
statistical precision. Thus, this has been the favored method in past experiments.
On the other hand, this method presents significant challenges. The maximum proton
kinetic energy is only ≈ 751 eV (cf. Eq. (2.46)), so decay protons must be accelerated to
much higher energy to be detected, typically to about (10 − 30) keV. Then, the proton
energy must be precisely measured prior to acceleration. This can be done by means of
an electrostatic filter or by time-of-flight (TOF) measurement. In both cases only one
component of the proton’s velocity is measured and its total energy must be inferred.
Experience has shown that controlling the measurement systematics presents by far the
greatest challenge, cf. Sec. 2.4.1. Amongst others, electric fields and space charges must
15A sign error in this formula was found by C. Habeck [78]. In the last line of Eq. (4.5) in Ref. [77] the
expression σ + x2 should be changed to σ − x2.
16The Coulomb correction F (Ee) to the electron energy spectrum is discussed in the following section.
2.1. THEORY OF NEUTRON BETA DECAY 15
Figure 2.3: The theoretical proton recoil spectrum Eq. (2.38) in neutron beta decay. Top: The
green line is the prediction from the SM with the recommended value for a = −0.103
[10], the black line shows how a deviation from that (a = 0) would look like. For
elucidation of the influence of the correlation a on the proton spectrum wp,[Nac68](Tp),
the red line shows the function g2(Tp) from Eq. (2.40). Bottom: The blue line shows
the ratio wp,[Nac68](Tp; a = −0.103)/wp,[Nac68](Tp; a = 0), to illustrate the dependence
of wp,[Nac68](Tp) on a.
be carefully controlled and proton scattering from residual gas molecules must be avoided,
in order not to distort the proton recoil spectrum (discussed in greater detail in Sec. 3.4
and Chap. 6).
For most of the investigations of systematic effects, the above description of the proton
recoil spectrum is sufficient. On the other hand, most of the Monte Carlo (MC) simulations
were performed in the infinite nucleon mass (INM) approximation17 [27] (see also the
following section), including the Coulomb correction to the electron energy spectrum.
For the analysis of measurement data, a more suitable description of the spectrum is
mandatory [74], denoted as
wp,C,α(Ep) = wp,C(Ep) · [1 + 0.01rC(y)] [1 + 0.01rρ + 0.01rp(y)] . (2.43)
This description comprises [74] (cf. Eqs. (3.1) to (3.12) in Ref. [74])
• the proton spectrum wp,C(Ep), including the F̂ (Ee, Ep) = 1 + παβ approximation
18,
• the higher-order Coulomb correction rC(y) to the proton energy spectrum,
• the model-independent order-α correction rρ = 1.505 to the total decay rate, and
• the radiative correction rp(y) to the proton energy spectrum,
17In the INM approximation mn →∞, mp →∞ limit, with mn−mp equal to the finite, true value. In
this case only the longitudinal momentum components (the pνsn, pesn, ppsn projections) are constrained
by the integration region, and we can integrate over φe and φν . As input for our MC simulations, we have
used mn = 1MeV/c2.
18In the F̂ (Ee, Ep) Coulomb correction, we take into account the proton energy dependence of the
proton recoil.
16 CHAPTER 2. NEUTRON BETA DECAY
with
Ep = mp + (Ep,max −mp)y. (2.44)
Here and in the following, we assume w.l.o.g. c = 1 and massless neutrinos (mν = 0).
Then,
∆2 −m2e [10]Ep,max = mp + = 938.272764(23)MeV2, (2.45)2mn
i.e.,
[10]
Tp,max = Ep,max −mp = 751(33)eV2. (2.46)
The corrections rC and rp are tabulated in Tables III and IV in Ref. [74], and are linearly
extrapolated to the proton energy regime.
To fit MC simulated data, we have used an approximation of wp,C(Ep) as given in
[74] (cf. Eq. (3.12) in Ref. [74]), only including the F̂ (Ee, Ep) approximation. The rela-
tive effect of this method on the correlation coefficient a is smaller than 0.1% (see also
Fig. 2.5b).
The Lepton Spectra
Using Fermi’s golden rule Eq. (2.8), the decay probability of neutrons is:
2π
dΓ(Ee) = |M 2fi| dρ~ e
(Ee), (2.47)
where the electron energy spectrum is given by
dρe(Ee) = we(Ee)dEe, (2.48)
with
(4π)2 √
we(Ee) = F (1, Ee) E2e −m2eEe (E0 − E )
2
e [1 + δR(Ee)] [1 +R0(Ee)] . (2.49)(2π~)6
Here,
def ∆2 −m2e [10]E0 = E 2e,max = ∆− = 1292.582(32)keV , (2.50)2mn
i.e.,
[10]
Te,max = Ee,max −me = 781.583(32)keV2. (2.51)
Equation (2.49) includes (cf., e.g., [80])
• the Coulomb correction by the Fermi function19 [70] (see also [81])
2πη ±ZαF (Z,Ee) = , with η = , (2.52)1− exp(−2πη) β
19Here Z is the atomic number of the daughter nucleus and the − sign refers to nuclear decays with
positron emission.
2.1. THEORY OF NEUTRON BETA DECAY 17
• the outer radiative cor[rection [73]
α mp 3
δR(Ee) = 3 ln( − )( )(2.53)2π me 4
arctanhβ − E0 − Ee − 3 2 (E0 − Ee)+4 ( 1 + lnβ ) 3Ee 2 me
4 2β
+ L
β 1 + β( )]
arctanhβ ( ) (E 2
+ 2 1 + 2 + 0
− Ee)
β − 4 arctanhβ ,
β 6E2e
with the Spen∫ce’s function L(z) = −Li2(z), where Li2 is the dilogarithm, i.e.:z ln |1− t|
L(z) = dt , (2.54)
0 t
• and the recoil correction [70[, 74] ( )
1 E E m2 E
R0(
e
Ee) = 2 + 2 10
e
λ − 2 e − 2 0 (2.55)
1 + 3λ2 mn ( mn mnEe mn)]
E 2
+ (1 + 2 ) −4 e m Eλ κ + 2 e + 2 0 ,
mn mnEe mn
where κ = f2 ≈ µp−µn′ 2 ≈ 1.85 [70] is the weak magnetism form factor.f1
In the INM approximation, we only include the Coulomb correction to the electron
energy spectrum, with the app(roximation of Eq.(2.52) [74]: )
2
F (1, Ee) ≈
πα 11 π
1 + + α2 − γE − ln (2βEeR) + , (2.56)
β 4 3β2
where
≈ 0.01R 1 fm ≈ and γE ≈ 0.5772. (2.57)4me
The electron energy spectrum in the INM approximation is shown in Fig. 2.4a.
In analogy to Eq. (2.48), the electron-antineutrino energy spectrum can be written as:
(4 √π)2
dρν(Eν) = (E
2 2 2
(2 ~)6 ν,max
+me − Eν) −me (Eν,max +me − Eν)EνdEν , (2.58)π
where
mp ∆2 +m2e [10]T 2ν,max ≡ Eν,max = ∆− me − = 782.008(33)keV . (2.59)
mn 2mn
The electron-antineutrino spectrum in the INM approximation is shown in Fig. 2.4b.
Figure 2.5a shows the influence of the Coulomb correction Eq. (2.56), in the INM
18 CHAPTER 2. NEUTRON BETA DECAY
(a) (b)
Figure 2.4: The theoretical (a) electron and (b) electron-antineutrino energy spectra in neutron
beta decay, in the INM approximation. The comparison between the black and the red
line shows the influence of the Coulomb correction Eq. (2.56) on the lepton spectra,
see also Fig. 2.5a. For details see the text. Input data for the MC simulation: Number
of generated events = 109 and a = −0.105 (derived from λ = −1.2701(25) [10]).
(a) (b)
Figure 2.5: Influence of several corrections on the proton and lepton energy spectra: (a) The ra-
tios w(T ;w/ F (1, Ee) from Eq. (2.56))/w(T ;w/o F (1, Ee)) show the influence of the
Coulomb correction Eq. (2.56), in the INM approximation, on the lepton and proton
energy spectra. (b) The ratios wp,C,α(Tp)/wp(Tp) show the influence of several approx-
imations and/or corrections on the proton recoil spectrum Eq. (2.43). The comparison
between the black, the red, and the blue line shows that the higher-order Coulomb
correction rC(y), the model-independent order-α correction rρ, and the radiative cor-
rection rp(y) have a rather small impact on the proton recoil spectrum, primarily at
high proton kinetic energies. For reference, the comparison between the black and
the green line shows the uncertainty resulting from the MC simulation. For details
see the text. Input data for the MC simulation (in INM approximation): Number of
generated events = 109 and a = −0.105 (derived from λ = −1.2701(25) [10]).
approximation, on the lepton and proton spectra. With the recommended value for λ =
−1.2701(25) [10], this is about a 1% effect. For reference, Fig. 2.5b shows the influence
2.1. THEORY OF NEUTRON BETA DECAY 19
of several approximations and corrections on the proton recoil spectrum Eq. (2.43). As
one can see, the higher-order Coulomb correction rC(y), the model-independent order-α
correction rρ, and the radiative correction rp(y) (for details see the previous section) have
a rather small impact on the proton recoil spectrum and primarly at higher proton kinetic
energies.
2.1.6 Extensions of the Standard Model
There are a number of extensions to the SM. The question, which theory is the right one,
can only be settled by experiments. Therefore, the search for physics beyond the SM is
one of the most active areas both in theoretical and experimental physics. Of particular
interest in the context of this thesis are the search for S and T interactions and for RH
currents.
The V−A description of neutron beta decay omits S and T interactions as described by
Eq. (2.9). Two possible models of these non-SM interactions can be tested using neutron
decay observables:
Left-Handed S and T Interactions
In the LH S and T model, non-vanishing Fierz interference terms b and bν appear. The
remaining free parameters are LV, LA, LS, and LT. The dependence of a, b, A, B0, bν ,
and C on these parameters follows from Eqs. (2.13) to (2.19), and (2.24):
L2 − L2 − L2 2 2 2 2 2
a = V A S
+ LT 1− λ − x + λ y
L2 + 3L2 + L2 + 3L2
= , (2.60)
2 2 2 2
V A S T 1 + 3λ + x + 3λ y
LVLS + 3LALT x+ 3λ2y
b = 2 2 2 2 2 = 2 , (2.61)L + 3L + L + 3L 1 + 3λ2 + x2 + 3λ2y2V A S T
L2 + L L − L L − L2 λ2 + λ− λxy − λ2y2
A = −2 A V A S T T2 = −2 , (2.62)LV + 3L2 + L2 + 3L2 2 2 2 2A S T 1 + 3λ + x + 3λ y
L2 − L L − L L + L2 λ2A V A S T T − λ− λxy + λ2y2B0 = 2 2 2 2 2 = 2 , and (2.63)LV + 3LA + LS + 3L 2 2 2 2T 1 + 3λ + x + 3λ y
2
bν = −
L L + L L − 2L L λy + x− 2λ y
2 V T A S A T2 = −2 , (2.64)LV + 3L2A + L2S + 3L2T 1 + 3λ2 + x2 + 3λ2y2
L L − L2 λ− λ2y2
C = 4 V Ax TC 2 = 4x , (2.65)LV + 3L
2 + L2 + 3 CL2 1 + 3λ2 2A S T + x + 3λ
2y2
where the neutron decay observables20 are expressed by three free parameters:
L
= A
LS L
λ , x = , and = Ty . (2.66)
LV LV LA
The direct determination of b through beta spectrum shape measurement is the most
sensitive way to constrain the size of the non-SM currents. The experiments discussed in
Sec. 2.2.1 measure the correlation coefficients from the electron spectra and asymmetries,
respectively. The published results on a, A, B, and C assume b = bν = 0. To make use
20Please note that, here, we omit the neutron lifetime τn, since otherwise we would have to determine
+ +
the possible influence of the Fierz term in Fermi decays, bF, on the Ft0 →0 values. A combined analysis
of neutron and superallowed 0+ → 0+ nuclear beta decays is found in Sec. 2.3.2.
20 CHAPTER 2. NEUTRON BETA DECAY
of measured values of a in a scenario involving a non-zero value for the Fierz term b, we
rewrite E(q. (2.13) for unpolar)ized neutron decay: ( )
d3 ∝ pΓ 1 + e · pν m
( 〈 〉)
a + eb ≈ a pe · p1 + νbme E−1e 1 + 〈 − 〉1 . (2.67)EeEν Ee 1 + bme Ee EeEν
The value quoted for a is then taken as a measurement of ā, defined through
a
ā = 〈 〉 . (2.68)
1 + bme E−1e
Here 〈·〉 denotes the weighted average over the part [Ee,1, Ee,2] of the beta spectrum
Eq. (2〈.49) o〉bserve∫d in the particular experimentEe,2
E−1
1
e = dEeF (1, Ee)pe (E0 − Ee)
2 [1 + δR(Ee)] [1 +R0(Ee)], (2.69)
f Ee,1
where the ∫statistical rate function f is given by:E0
f = dEewe(Ee). (2.70)
me
This procedure has been also applied in Refs. [17, 27, 76]. Reported experimental values
of A, B, and C are interpreted as measureme〈nts o〈 〉 〈 − 〉
f
A B 10 + bνme Ee 〉 −x (A+B〈 ′= = = C 0)− x〉CbνĀ
1 + bm E−1
, B̄ −1 , C̄ , (2.71)
e e 1 + bme Ee 1 + bm −1e Ee
assuming integration over all electrons. This procedure is not perfect. The presence of a
Fierz term b might infl〈uenc〉e systematic uncertainties. For example, the background esti-mate in PERKEO II assumes the SM dependence of the measured count rate asymmetry
on Ee. The term m E−1e e depends on the part of the electron spectrum used in each
experiment.
Right-Handed S and T Interactions
In the RH S and T model, the Fierz terms b and bν are zero [17]. The remaining free
parameters are LV, LA, RS, and RT. The dependence of a, A, B, C, and τn on these
parameters follows from Eqs. (2.13) to (2.17), (2.24), and (2.32) and (2.34):
L2 2 2 2V − LA −RS +RT 1− λ2 − x2 + λ2y2a = 2 = , (2.72)LV + 3L2 +R2A S + 3R2T 1 + 3λ2 + x2 + 3λ2y2
L2 + L 2 2 2 2
= −2 A V
LA +RSRT +RT − λ + λ+ λxy + λ yA 2 2 2 2 = 2 , (2.73)LV + 3LA +RS + 3RT 1 + 3λ2 + x2 + 3λ2y2
L2A − LVL 2 2A +RSRT −RT λ − λ+ λxy − λ2y2B = 2 2 = 2 , (2.74)L + 3L2 +R2 + 3R2 1 + 3λ2 + x2 + 3λ2 2V A S T y
LVLA +R2 2 2
C = 4x T
λ+ λ y
C
L2 + 3L2 +R2 2
= 4xC , and (2.75)+ 3R 1 + 3λ2 + x2 + 3λ2y2V A S T
Ft0+→0+ + +2(L2V +R2S) Ft0 →0 2(1 + x2)τn = 2 2 2 2 = , (2.76)f ln (2) L + 3L +R + 3R f ln (2) 1 + 3λ2 + x2 2 2R V A S T R + 3λ y
where the neutron decay observables are expressed by three free parameters:
L
= A
R R
λ , x = S , and y = T . (2.77)
LV LV LA
2.1. THEORY OF NEUTRON BETA DECAY 21
Left-Right Symmetric Models
An electrically charged gauge boson outside the SM is generically denoted W ′. The most
attractive candidate forW ′ is theWR gauge boson associated with the left-right symmetric
models [82, 83], which seek to provide a spontaneous origin for parity violation in weak
interactions. WL and WR may mix due to spontaneous symmetry breaking. The physical
mass eigenstates are denoted as
W1 = WL cos ζ −WR sin ζ and (2.78)
W2 = WL sin ζ +WR cos ζ, (2.79)
where W1 is the familiar W boson and ζ is the mixing angle between the two mass
eigenstates. In the manifest left-right symmetric (MLRS) model, there are only three
free parameters, the mass ratio δ = m21/m22, ζ, and λ′. Here, m1 = 80.398(23)GeV/c2
[10] and m2 denote the masses of W1 and W2, respectively. Since LV = 1 (CVC) and
LS = LT = RS = RT = 0, the coupling constants LA, RV, and RA depend on δ, ζ, and λ′
as described in Refs. [27, 84, 85]21,22:
′ · (1 + tan ζ)− δ tan ζ(1− tan ζ)LA = λ = λ ≈ λ(1 + 2ζ), (2.80)(1− tan ζ) + δ tan ζ(1 + tan ζ)
δ(1 + tan ζ)− tan ζ(1− tan ζ)
RV = ≈ δ − ζ, and (2.81)(1− tan ζ) + δ tan ζ(1 + tan ζ)
′ · δ(1− tan ζ) + tan ζ(1 + tan ζ)RA = λ ≈ λ(1 + 2ζ)(δ + ζ). (2.82)(1 + tan ζ)− δ tan ζ(1− tan ζ)
The dependence of a, A, B, C, and τn on δ, ζ, and λ′ follows from their respective
dependence on LV, LA, RV, and RA:
L2V − L2A +R2 2 ′
2 2 2
a = V
−RA 1− λ +RV −RA
L2
= , (2.83)
V + 3L
2
A +R
2 2
V + 3RA 1 + 3λ′
2 +R2 2V + 3RA
L2 2A + LVLA −RA −RVR ′
2
A λ + λ′ −R2
A = −2 = −2 A
−RVRA
L2 2 2 2
, (2.84)
V + 3LA +RV + 3RA 1 + 3λ′
2 +R2V + 3R
2
A
L2A − L 2VLA −RA +RVRA λ′
2 − λ′ −R2 +R R
= 2 A V AB
L2 + 3L2
= 2 , (2.85)
V A +R
2
V + 3R
2 ′2 2
A 1 + 3λ +RV + 3R
2
A
L L −R R λ′V A V A −R R
C = 4 V AxC 2 2 2 2 = 4xC ′2 , and (2.86)LV + 3L 2 2A +RV + 3RA 1 + 3λ +RV + 3RA
F 0+→0+ + +t 2(L2 +R2 ) Ft0 →0 2(1 +R2V V V)τn = · = · .
fR ln (2) L2V + 3L
2
A +R
2 2 2 2 2
V + 3RA fR ln (2) 1 + 3λ′ +RV + 3RA
(2.87)
So far, there is no experimental evidence of a WR weak gauge boson. Hence WR and
W2 are expected to be very massive particles. In the case of WR couplings to RH quarks,
the limit MW > 2.5TeV from KL −KS mixing [87] is severe23. The SM is included inR
21A sign error in RV in Ref. [85] was found by H. Mest [32]: In the numerator of Eq. (2.81) the expression
tan ζ(1 + tan ζ) should be changed to tan ζ(1− tan ζ).
22Please note that we, such as [27], express RA with a prefactor λ′ compared to Refs. [32, 85, 86]. This
retains the definition LV = 1 (CVC).
23There are also WR contributions to the neutron electric dipole moment (nEDM). The combined
constraints from KL decay and the nEDM (dn < 2.9×10−26 e cm [88]) yield a bound ofMWR > (2−6)TeV
[87].
22 CHAPTER 2. NEUTRON BETA DECAY
the MLRS model when no mixing occurs (ζ = 0) and the W2 boson is infinitely heavy.
Then, δ = 0 and λ = λ′.
The experimentally determined values for the angular correlation coefficients and the
neutron lifetime can be used to test the SM as well to search for evidence of possible exten-
sions to it. This is done in the following sections, applying the results of the measurements
introduced in Sec. 2.2.1.
2.2 Tests of the Standard Model
As discussed earlier in Chap. 1 the angular correlation coefficients overdetermine the V−A
description of neutron beta decay. Therefore, one can test the validity and consistency of
its SM description. Before doing so, we have to analyze the currently available data on
neutron decay.
2.2.1 Experimental Data
In Secs. 2.2 and 2.3, we present results of least-squares fits, using recent experimental data
as well as target uncertainties for planned experiments on neutron decay. The various
observables Oi, for i = 1, . . . , N , depend non-linearly on a set of M parameters pi, for
i = 1, . . . ,M . Given the theoretical expressions of the observables Θi(p) one defines the
figure-of-merit [function χ
2 w]hich is minimized to determine the best-fit parameters by∑N
2 O
2
= i
−Θi(p)
χ , (2.88)
σ
i=1 i
where Oi is the measured value and σi is the corresponding (1σ) experimental error. The
fit parameters p are defined below as |Vud|, ratios of the different couplings Lj and Rj
respectively the MLRS couplings δ, ζ, and λ′. The principle of non-linear χ2 minimization
is discussed, e.g., in Ref. [89]. Figures 2.6a to 2.10b show the present and expected
future limits from neutron decay, respectively. The confidence regions in 2 dimensions, or
confidence intervals in 1 dimension, are defined as in Ref. [90].
Present Limits
We first analyze the presently available data on neutron decay. As input for our study we
used:
a = −0.103(4) and B = 0.9807(30), (2.89)
both from Ref. [10], as well as
F + +t0 →0 = 3071.81(83) s (2.90)
as the average value for superallowed 0+ → 0+ nuclear beta decays, the so-called superal-
lowed Fermi transitions (SAF) (from Ref. [76]). We used our own averages for τn and A,
as follows.
2.2. TESTS OF THE STANDARD MODEL 23
The most recent results24 of Serebrov et al. [93], τn = 878.5(8) s, and Pichlmaier et al.
[94], τn = (880.7 ± 1.8) s, were not included in the PDG 2010 average [95]. We preferred
not to exclude the measurement by Serebrov et al. without being convinced that it is
wrong, and include it in our average to obtain
τn = (881.8± 1.4)s. (2.91)
Our average includes a scale factor of 2.5, as we obtain χ2 = 45 for 7 degrees of freedom.
The statistical probability for such a high χ2 is 1.5 × 10−7. If our average were the true
value of the neutron lifetime τn, both the result of Serebrov et al. and the PDG 2010
average would deviate at the (2− 3)σ level. We note that in the 2011 partial update web
version of the PDG 2010 review [10], a new average25 is given, τn = (881.5 ± 1.5) s, in
close agreement with our average Eq. (2.91).
Two beta asymmetry experiments have completed their analyses since the PDG 2010
review, namely UCNA [29] and PERKEO II [30]26. The UCNA collaboration has pub-
lished A = −0.11966(89)+0.00123 27−0.00140 [29] . The last PERKEO II run has yielded a preliminary
value of A = −0.1198(5) [97]. We include these two results in our average, and obtain
A = −0.1186(9), (2.92)
which includes a scale factor of 2.3 based on χ2 = 28 for 5 degrees of freedom. The
statistical probability for such a high χ2 is 5× 10−5, not much better than in the case of
τn. We note that in the 2011 partial update web version of the PDG 2010 review, a new
average is given, A = −0.1176(11), in agreement with our average Eq. 2.92.
Hence, we find that the relative errors are about 4% in a, 0.9% in A, and 0.3% in
B. We will not use C = −0.2377(26) [10] in the analysis of present results, since the
PERKEO II results for B and C are derived from the same data set.
Future Limits
About a dozen new instruments are currently planned or under construction. For recent
reviews see Refs. [18, 19, 22, 23]. We will discuss a future scenario which assumes the
following improvements in precision in a couple of years.
• ∆a/a = 0.1%: Measurements of the neutrino-electron correlation coefficient a with
the aSPECT, aCORN [98], Nab [99], and PERC experiments are projected or un-
derway (for details see Sec. 2.4.2 and App. C).
• ∆b = 3× 10−3: Measurements of the Fierz interference term b in neutron decay are
planned by the Nab [99], UCNb [100], and PERC collaborations.
• ∆A/A = 3 × 10−4: Measurements of the beta asymmetry parameter A with
PERKEO III [32, 101], UCNA [102], abBA [103], and PERC [104] are almost
analyzed, planned, or underway.
24We note that a recent experiment [91] measured the neutron lifetime with a magneto-gravitational
trap. Ezhov reported [92] at the 7th International Workshop “Ultracold and Cold Neutrons. Physics and
Sources.” a neutron lifetime of τn = (878.2 ± 1.9) s, which is in excellent agreement with the result of
Serebrov et al. We disregard the result, as it is not published yet.
25The new PDG 2011 average includes the result of Serebrov et al., as Pichlmaier et al. obtained a
value closer to the value of Serebrov et al..
26Publication of the result of the last PERKEO II run [31] is underway.
27In the analysis of present results we will use the previously published value A = −0.1138(46)(21) [96].
24 CHAPTER 2. NEUTRON BETA DECAY
• ∆B/B = 0.1%: The abBA [103] and UCNB [105] collaborations intend to measure
the neutrino asymmetry parameter B. PERC is also exploring a measurement of B.
• ∆C/C = 0.1%: The aSPECT [48] and PANDA [106] collaborations plan measure-
ments of the proton asymmetry parameter C; PERC may follow suit as well.
• ∆τn = 0.8 s: Measurements of the neutron lifetime τn with beam experiments [107,
108], material bottles [109, 110], and magnetic storage experiments [91, 111–114] are
planned or underway.
Our assumptions about future uncertainties for a, A, B, and C reflect the goal accuracies
in the proposals, while for τn we only assume the present discrepancy to be resolved. Our
assumed ∆τn corresponds to the best uncertainty claimed in a previous experiment [93].
Our scenario “future limits” assumes that the SM holds and connects the different
observables. We used a = −0.10588, b = 0, B = 0.98728, C = −0.23875, and τn = 882.2 s
derived from + +A = −0.1186 and Ft0 →0 = 3071.81 s. These values agree with the present
measurements within 2σ.
2.2.2 Test of the V−A Description
A useful model-independent test of the self-consistency of the SM is derived in Ref. [115].
Based on the expressions for a, A, and B from Eq. (1.4), one can construct two equations:
def
F1 = 1 +A−B − a ≡ 0 and (2.93)
def
F2 = aB −A−A2 ≡ 0 . (2.94)
The PDG 2011 review for the correlation coefficients [10] yields:
F1 = 0.0047(51) and F2 = 0.0028(40), (2.95)
where the uncert√ainties in F1 and F2
∆F1 = √(∆a)2 + (∆A)2 + (∆B)2 and (2.96)
∆F2 = (B∆a)
2 + ((1 + 2A) ∆A)2 + (a∆B)2 (2.97)
are dominated by the poor knowledge of a. From our present limits in Sec. 2.2.1 we derive
an improved test of the self-consistency of the SM:
F1 = 0.0037(51) and F2 = 0.0035(40). (2.98)
Our values are consistent with the V−A theory, but even a small improvement in a would
considerably improve the precision of this comparison. New neutron decay experiments
(cf. Sec. 2.2.1) could lead to accuracies of
∆F1 = 0.001 and ∆F2 = 0.00015, (2.99)
which would make this comparison much more useful.
2.2. TESTS OF THE STANDARD MODEL 25
(a) (b)
Figure 2.6: Determination of |Vud| and λ = LA/LV: (a) Present limits from a, A, B, and τn
in neutron decay. (b) Future limits from neutron decay, assuming improved and
independent measurements of a, A, B, C, and τn. Analogous limits for |Vud| from
superallowed 0+ → 0+ nuclear beta decays, pion decays, and T = 1/2 mirror decays
are indicated. For details see the text. All bars correspond to single parameter limits.
2.2.3 Unitarity of the CKM Matrix and Determination of |Vud| and λ
At present, the most accurate value of |Vud| is derived from measurements of superallowed
0+ → 0+ nuclear beta decays (SAF) [10, 75]:
|Vud| = 0.97425(22). (2.100)
Combined with measurements of |Vus| and |Vub| from kaon decays and semileptonic decay
of B-mesons [10], respectively, this leads to the currently most precise test of the unitarity
of the CKM matrix [10, 36, 75]:
∆ d=ef 1− |Vud|2 − |Vus|2 − |Vub|2 = (1± 6)× 10−4. (2.101)
However, the extraction of Vud involves calculations of radiative and nuclear structure
corrections for the Fermi transition in nuclei. Even though these calculations have been
done with high precision [76, 116] (and references therein), questions concerning these
corrections have been raised [117–120]. In view of the intrinsic theoretical complexity of
nuclear beta decays, it is highly desirable to derive the limits from neutron beta decay, with
at least comparable precision, as they are independent of nuclear structure. Unfortunately,
a disturbing inconsistency persists within the neutron decay data (cf. [10], Fig. 1.1, and
Sec. 2.2.1).
Figure 2.6a shows the present limits from neutron decay. Free parameters |Vud| and
λ = LA/LV where fitted to the observables a, A, B, τn. Additionally, to take into account
uncertainties in radiative corrections, we fitted the denominator of Eq. (2.36) to a ‘data
point’ (4908.7± 1.9) s. The contours around the minimum correspond to the three levels
of constant χ2: χ2min + 2.3, χ
2
min + 4.61, and χ
2
min + 6.17, respectively, where χ
2
min is the
value of χ2 at the minimum. The allowed regions for |Vud| and λ, to 68.3%, 90%, and
26 CHAPTER 2. NEUTRON BETA DECAY
95.4% C.L., are obtained by varying all three parameters around the minimum. At the
1σ confidence level, we find
|Vud| = 0.9745(17) and λ = −1.2731(24). (2.102)
Hence, |Vud| is in agreement with, but less accurate than SAF beta decays. The confidence
interval for a single parameter is obtained by varying the values of the other two parameters
around the minimum. Here, the 1σ and 2σ confidence intervals correspond to the two
levels of constant χ2: χ2 2 2min + 1 and χmin + 2 , respectively. Along with |Vus| = 0.2252(9)
and |Vub| = 3.89(44) × 10−3 from the PDG 2011 review [10], our test of the unitarity of
the CKM matrix yields:
∆ = (−4± 33)× 10−4, (2.103)
in good agreement with the SM expectation, but also less sensitive than SAF beta decays.
Figure 2.6b demonstrates the sensitivity of our future scenario. New neutron decay
experiments could considerably improve the accuracies of |Vud| and λ to
∆|Vud| = 0.00049 and ∆λ = 0.00011, (2.104)
both at the 1σ confidence level. To further improve the accuracy of |Vud| (and ∆) to, e.g.,
|Vud| = 0.00021, competitive to SAF beta decays, one would have to measure the neutron
lifetime τn with a precision of ∆τn = 0.1 s. Nevertheless, the precision in |Vud| extracted
from neutron decay experiments would mainly be limited by radiative corrections, similar
to the extraction of |Vud| from SAF beta decays. In particular, an update of the theoretical
corrections would be required.
Recent studies of T = 1/2 nuclear mirror transitions [121] have reached a precision in
|Vud| competitive to neutron decays, as can be seen from Fig 2.6a. In Figs. 2.6 we also
show the limits from pion beta decay measurements [122], namely from the rare decay
mode π+ → π0 + e+ + νe. As a pure vector transition, theoretical uncertainties in the
determination of |Vud| from pion beta decay are very small. The difficulty in measuring the
pion decay rate is due to the small branching ratio of O(10−8). Higher counting statistics
would be required to make this approach competitive with SAF beta decays.
2.3 Searches for Physics Beyond the Standard Model
Our fits are not conclusive if all 8 coupling constants Lj and Rj , for j =V,A,S,T, are
treated as free parameters. We are more interested in restricted analyses presented below.
Experiments quote a, A, B, and C after applying (small) theoretical corrections for recoil
and radiative effects; we neglect any dependence on non-SM physics in these corrections.
We note that in Ref. [49], citing this thesis, the influence of improved measurements
of the neutrino-electron correlation coefficient a on non-SM physics is extremely over-
estimated. The figures given in Ref. [49] reflect how much the area of the 95% contours
could change due to an improved measurement of a. Hence, these figures are inconclusive.
We further note that limits similar to those presented below were derived by Dubbers
and Schmidt [19], using a slightly different neutron and nuclear data set. First, Dubbers
and Schmidt included the proton asymmetry parameter C in their analysis of present
results, by using for the neutrino asymmetry parameter B only the value of Serebrov et
al. [123, 124]. Secondly, in the search for LH S and T currents, they included the ratio
2.3. SEARCHES FOR PHYSICS BEYOND THE STANDARD MODEL 27
(a) (b)
Figure 2.7: Left-handed scalar and tensor currents: (a) Present limits from neutron decay (only
a, A, and B). The SM values are at the origin (white) of the plot. Analogous limits
extracted from muon decays are indicated. (b) Future limits from neutron decay,
assuming improved and independent measurements of a, b, A, B, and C. Analogous
limits extracted from muon decays are not indicated since they exceed the scale of the
plot. Other limits are discussed in the text. All bars correspond to single parameter
limits.
rFt in their analysis, using the average F 0
+→0+t value for SAF decays (see Sec. 2.3.2 for
details). Additionally, they also derived limits for the Fierz interference terms b and bν ,
both consistent with the SM prediction.
2.3.1 Constraints from Neutron Decay Alone
Left-Handed Scalar and Tensor Currents
Addition of LH S and T currents to the SM leaves LV = 1, LA = λ, LS, and LT as the non-
vanishing parameters. Non-zero Fierz terms b and bν appear in this model. Figure 2.7a
shows the current limits from neutron decay.
Free parameters λ, LS/LV, and LT/LA were fitted to the observables ā, Ā, B̄, and b
and C̄ in case 〈of our〉futur〈e scen〉ario (see Sec. 2.1.6 for d〈etails)〉. We have used the followingvalues for m E−1e e in our study (cf. Sec. 2.1.6): m −1e Ee = 0.5393 for Ā, dominated
by PERKEO II [43], m E−1e e = 0.6108 for B̄〈, dom〉 inated by Serebrov et al. [123, 124]
and PERKEO II [125], and the mean value m E−1e e = 0.6556, taken over the whole beta
spectrum, for ā and C̄. The 68.3% C.L.s are
λ = −1.322(48), LS/LV = 0.099(88), and LT/LA = 0.056(138). (2.105)
Unlike the following sections, here we omit the neutron lifetime τn, since otherwise we
would have to determine the possible influence of the Fierz term in Fermi decays, bF, on
the Ft0+→0+ values. A combined analysis of neutron and SAF beta decays is found in
Sec. 2.3.2.
28 CHAPTER 2. NEUTRON BETA DECAY
Figure 2.7b presents the impact of projected measurements in our future scenario. For
comparison, a recent combined analysis of nuclear and neutron physics data [17] finds
LS/LV = 0.0013(13) and LT/LA = 0.0036(33), (2.106)
with 1σ statistical errors. It includes the determination of the Fierz term bF from su-
perallowed beta decays, updated in Ref. [76], which sets a limit on LS that is hard to
improve with neutron beta decay alone. As in the recent survey of Severijns et al. [17], we
do not include the limits on tensor couplings obtained [126] from a measurement of the
Fierz term bGT in the forbidden Gamow-Teller decay of 22Na, due to its large log ft(=7.5)
value. Neutron decay has the potential to improve the best remaining nuclear limit on
LT as provided by a measurement of the longitudinal polarization of positrons emitted
by polarized 107In nuclei (log ft = 5.6) [127, 128]28,29. Limits from neutron beta decay
are independent of nuclear structure. The stringent limit on LT in the combined analysis
of nuclear and neutron decay data [17] stems mainly from measurements of τn and B in
neutron beta decay. New neutron decay experiments alone could lead to uncertainties of
∆(LT/LA) = 0.0023 and ∆(LS/LV) = 0.0083, (2.107)
both at the 1σ confidence level. Then, the precision in ∆(LT/LA) would be competitive
with the combined analysis of neutron and nuclear physics data [17]. We note that super-
symmetric (SUSY) contributions to the SM can be discovered at this level of precision, as
discussed in Ref. [131].
Right-Handed Scalar and Tensor Currents
Adding the RH S and T currents to the SM yields LV = 1, LA = λ, RS, and RT as the
remaining non-zero parameters. The observables depend only quadratically on RS and
RT, i.e., the possible limits are less sensitive than those obtained for LH S, T currents.
Figure 2.8a shows the present limits from neutron beta decay. A similar analysis of this
scenario was recently published in Ref. [132] (see also Ref. [30]).
Free parameters λ, RS/LV, and RT/LA were fitted to the observables a, A, B, τn,
and C, in case of our future scenario. Additionally, to take into account uncertainties
in the Ft values and in radiative corrections, we fitted F +t0 →0+ and fR to ‘data points’
3071.81(83) s and 1.71385(34), respectively. The 68.3% C.L.s are
λ = −1.2727(37), RS/LV = 0.000(87), and RT/LA = 0.000(78). (2.108)
The Fierz interference terms b and bν are zero in this model. Hence, measurements of
b (or bF in SAF beta decays) can invalidate the model, but not determine its parameters.
Figure 2.8b shows the projected improvement in our future scenario. The gray ellipse
stems from a recent survey of the state of the art in nuclear and neutron beta decays [17].
28Please note that in Figs. 1 and 2 in Ref. [6] the limit from Ref. [127, 128] was by mistake shifted to
higher LT/LA values.
29We note that recent experiments [129, 130] measured the beta asymmetry parameter A in Gamow-
Teller decays of polarized 114In (log ft = 4.5) and 60Co nuclei (log ft = 7.5). Wauters et al. reported
limits on possible T currents and WR bosons:
• 114In [129]: LT/L 2A = +0.029(55) and m2 > 230GeV/c (both 90% C.L.),
• 60Co [130]: LT/LA = −0.038(28) and m2 > 245GeV/c2 (both 90% C.L.).
We do not show these limits, as they are inferior to the limits presented in Ref. [127, 128].
2.3. SEARCHES FOR PHYSICS BEYOND THE STANDARD MODEL 29
(a) (b)
Figure 2.8: Limits on right-handed scalar and tensor currents: (a) Current limits from a, A, B,
and τn in neutron decay. (b) Projected future limits from neutron decay, assuming
improved measurements of a, A, B, C, and τn. The SM prediction is at plot origin
(white). As a comparison, we show limits from a survey of nuclear and neutron beta
decays [17], and limits from muon decays and neutrino mass measurements. The gray
ellipse is the present 86.5% contour from Ref. [17]. The muon limit on RS/LV is larger
than the scale of the plot. For details see the text.
New neutron decay experiments alone could considerably improve the limits on RH S and
T currents, to
∆(RS/LV) = 0.0275 and ∆(RT/LA) = 0.0173, (2.109)
with 1σ statistical error.
Hypothetical W ′ Bosons
Addition of RH V and A currents to the SM leaves δ, ζ, and λ′ as the non-vanishing
parameters. Figure 2.9a shows the current limits from neutron beta decay. The fit pa-
rameters δ, ζ, ′, F 0+λ t →0+ , and fR, were fitted to the observables a, A, B, τn, and C, in
case of our future scenario. At the 1σ confidence level, we find
λ′ = −1.2916(87), ζ = −0.080(84),
δ < 0.071, and hence m2 > 302GeV/c2. (2.110)
Measurements of the polarized observables, i.e., the electron, neutrino, or proton asym-
metries, lead to important restrictions, but are at present inferior to limits on the mixing
angle ζ from µ decays [134]. They are also inferior to limits on the mass m2 from direct
searches for extra W bosons [10]. Comparison of beta decay limits with high energy data
is possible in our minimal MLRS model. For example, the comparison with W ′ searches
at Tevatron [136] assumes a RH CKM matrix identical to the LH one and identical cou-
plings. In more general scenarios the limits are complementary to each other since they
probe different combinations of the RH parameters [137].
30 CHAPTER 2. NEUTRON BETA DECAY
(a) (b)
Figure 2.9: Limits on hypothetical W ′ bosons: (a) Present limits from a, A, B, and τn in neutron
decay. (b) Future limits from neutron decay, assuming improved measurements of a, A,
B, C, and τn. The SM values are at the origin (white) of the plot. As a comparison,
we show analogous limits from nuclear decays [127, 128], muon decays [133, 134],
lepton scattering (deep inelastic ν-hadron, ν-e scattering, and e-hadron interactions)
[135], and a direct search at D
0 [136]. The value of |Vud| from superallowed 0+ → 0+
nuclear beta decays was used to set a limit on ζ, assuming that the CKM matrix for
left-handed quarks is strictly unitary (see Ref. [76]). Limits from kaon and B-meson
mixing and the neutron EDM (MWR > (2− 6)TeV/c2) [87] are not shown.
Figure 2.9b presents the improvement from planned measurements in our future sce-
nario. The χ2 minimization converges to a single minimum at mass m2 =∞; with χ2 = 0,
i.e., the mixing angle ζ is not defined at this minimum. The 68.3% C.L.s are
δ < 0.0196, which yields m2 > 574GeV/c2. (2.111)
In the mass range > 1TeV, not excluded by collider experiments, we would improve the
limit on ζ from µ decays slightly.
We emphasize that all presented RH coupling limits (RS, RT, δ, and ζ) assume that the
RH (Majorana) neutrinos are light (m 1MeV). The RH interactions are kinematically
weakened by the masses of the predominantly RH neutrinos, if these masses are not much
smaller than the electron endpoint energy in neutron decay (782 keV). If both theW boson
and neutrino left-right mixing angles were zero, and if the RH neutrino masses were above
782 keV, RH corrections to neutron decay observables would be completely absent.
2.3.2 Left-Handed Coupling Constraints from a Combined Analysis
of Neutron and Nuclear Decays
Adding the LH S and T currents to the SM yields LV = 1, LA = λ, LS, and LT as
the remaining non-zero parameters. As discussed earlier, non-vanishing Fierz interference
terms b and bν appear in this model. Unlike Sec. 2.3.1 and Figs. 2.7, here, we include the
neutron lifetime τn in our analysis, as the neutron lifetime is sensitive to the Fierz term,
and the comparison of different nuclei shows that the effect of on F 0+L t →0+S is very
small.
2.3. SEARCHES FOR PHYSICS BEYOND THE STANDARD MODEL 31
To make use of measured values of τn in a scenario involving a non-zero value for the
Fierz term b, we rewrite Eqs. (2.32) and (2.34) in analogy to Eq. (2.68)30,31
F +→ + F
0+→0+t
t0 0 = − and (2.115)1 + b γ 〈W 1F 〉
τn
τn = 〈 − 〉1 , (2.116)1 + bme Ee
where [27, 76] (see also [75])
F +ξ t0 →0+ KF = ( ) and (2.117)
G2F|V |2ud 1 + ∆VR
K
ξτn = ( ) , (2.118)
fR ln (2)G2F|V 2ud| 1 + ∆VR
with32,33
ξFbF = √±4<(LSL∗V +R R∗S V), (2.119)
γ = 1− (αZ)2, and (2.120)
K 2π~ ln 2
= = 8120.2787(11)× 10−10 GeV−4 s. (2.121)
(~c)6 (m c2e )5
Here bF is the Fierz term in Fermi decays. For ξ and ξF see Eqs. (2.13) and (2.35),
30Please note that, here, Ft0+→0+ does not denote the average value for SAF beta decays from Ref. [76].
31In Eq. (2.122) 〈·〉 denotes the weighted average over the part of the electron/positron spectrum
observed in the par∫ticular experiment [138] (see also Eq. (2.69))〈 W0
W−1
〉 1
= dWF (±Z,W )S(±Z,W )p(W −W )20 , (2.112)
f̃ 1
with the sta∫tistical rate function f̃W0
f̃ = dWF (±Z,W )S(±Z,W )pW (W0 −W )2, (2.113)
1
where, to appropriate order (αZ)3 fo(r our present concern, [70] )
2
F (±Z,W ) ≈ 1± π 11 παZ +( − γE − ln (2pR) + (αZ)
2
β 4 ) 3β2
± 11πβ − γ 3E − ln (2pR) (αZ) . (2.114)
4
Here, W = E/m√e is the total electron/positron energy E in 〈electro〉n rest-mass units, W0 its maximum
value, and p = W 2 − 1 its momentum. In calculating the γ W−1 values, we put the shape-correction
function S(±Z,W ) to unity (compare Ref. [116]).
32The − sign in Eq. (2.119) refers to nuclear decays with positron emission.
33Compared with Ref. [74], here, the factor 4 in Eq. (2.119) considers the nuclear matrix elementMF
(see also Fn. 14 and Ref. [76]).
32 CHAPTER 2. NEUTRON BETA DECAY
〈 〉
Table 2.1: Ft values of superallowed beta decays [76] used in our study. The γ W−1 values have
been derived from the decay transition energies, QEC, summarized in [76].
Parent nucleus, T = −1 10C 14O 22Mg 34z Ar
Ft [s] 3076.7± 4.6 3071.5± 3.3 3078.0± 7.4 3069.6± 8.5
Q〈 2EC 〉 [keV/c ] 1907.87(11) 2831.24(23) 4124.55(28) 6062.98(48)
γ W−1 0.6185 0.4375 0.3072 0.2106
Parent nucleus, Tz = 0 26Alm 34Cl 38Km 42Sc
Ft [s] 3072.4± 1.4 3070.6± 2.1 3072.5± 2.4 3072.4± 2.7
Q〈EC 〉 [keV/c2] 4232.66(12) 5491.64(23) 6044.40(11) 6426.28(30)
γ W−1 0.2993 0.2321 0.2110 0.1982
46V 50Mn 54Co 62Ga 74Rb
3073.3± 2.7 3070.9± 2.8 3069.9± 3.3 3071.5± 7.2 3078± 13
7052.49(16) 7634.45(7) 8244.37(28) 9181.07(54) 10417.3± 4.4
0.1805 0.1665 0.1538 0.1373 0.1192
respectively. This results in
Ft0+→ K 10+ = ( ) · ( ) (2.122)
G2F|Vud|2 1 + ∆VR 2 L2 2 −1( ) V
+ LS ± 2LVLSγ 〈W 〉
K 1
= · and
G2 |V |2 1 + ∆Vud 2 (1 + x2 ± 2xγ 〈W−1〉)F R
K
τn = ( ) (2.123)
f ln (2)G2R F|V |2 1 + ∆Vud R
× 1 〈 〉
L2 2 2V + 3LA + LS + 3L
2
T + 2 (LVLS + 3LAL )m E
−1
T e e
K
= ( )
f ln (2)G2 |V 2 VR F ud| 1 + ∆R
× 1 〈 〉 .
1 + 3λ2 + x2 + 3λ2y2 + 2 (x+ 3λ2y)m E−1e e
Eq. (2.122) considers the possible influence of the Fierz term b on the F +t0 →0+F 〈values〉.
This method is not perfect either. We emphasize that the de〈nomin〉ator 1 + b γ W−1F
is different for every nuclei. In our study,〈we ha〉ve used for γ W
−1 the values given in
Table 2.1, where the rows give the measured pa〈rent n〉ucleus, the experimental F
+ +
t0 →0
values, and the derived mean values for γ W−1 . We used the experimental data of the
13 most precise results of Ref. [76] only. The γ W−1 values are derived from the decay
transition energies, QEC. For nuclear decays with positron emission34, the energy W0
relates to the QEC values listed in Table 2.1 via [139]:
QEC
W0 = − 1. (2.124)
me
34Currently, 13 transitions with positron emission, ranging from 10C to 74Rb, have been measured with
high precision.
2.3. SEARCHES FOR PHYSICS BEYOND THE STANDARD MODEL 33
(a) (b)
Figure 2.10: Left-handed scalar and tensor currents: (a) Current limits (only a, A, B, and τn).
(b) Projected future limits, assuming improved measurements of a, b, A, B, C,
and τn. The SM prediction is at plot origin (white). Other limits are discussed in
Sec. 2.3.1. All bars correspond to single parameter limits. Please note the different
scale compared to Fig. 2.7.
We note that Eqs. (2.122) and (2.123) both depend on the element Vud of the CKM matrix.
Hence, we obtain |Vud| as further non-vanishing parameter in this model.
Figure 2.10a shows the present limits. Free parameters λ, LS/LV, LT/LA, and |Vud|
were fitted to the observables ā, Ā, B̄, τn, Ft0+→0+ (for 10C to 74Rb), and b and C̄, in
case of our future scenario. Additionally35, to take into account uncertainties in radiative
corrections, we fitted ∆VR and fR to ‘data points’ 2.361(38)% and 1.71385(34). We find
λ = −1.2729(18), |Vud| = 0.9745(4),
LS/LV = 0.0010(14), and LT/LA = 0.0002(25), (2.125)
with 1σ statistical error.
Figure 2.10b presents the impact of projected measurements in our future scenario.
New neutron decay experiments, combined with SAF decay measurements, could lead to
uncertainties of
∆(LS/LV) = 0.0013 and ∆(LT/LA) = 0.0009, (2.126)
both at the 1σ confidence level. Hence, neutron beta decay has the potential to
considerably improve the limit on LH T currents as provided by the recent survey of
Severijns et al. [17].
In summary, new physics may be within reach of precision measurements in neutron
beta decay in the near future.
2.3.3 Limits from Other Fields
The search for physics beyond the SM is one of the most active areas in experimental
physics. Here, we only give an overview of experiments which also search for RH currents
or for S and T interactions. For a collection of recent results see Ref. [140].
35Here, we omit the uncertainties in the physical constants ~, c, me, and GF/(~c)2.
34 CHAPTER 2. NEUTRON BETA DECAY
Constraints from Muon and Pion Decays
Muon decay provides arguably the theoretically cleanest limits on non-(V−A) weak inter-
action couplings [10, 141]. Muon decay involves operators that are different from the ones
encountered in neutron, and generally hadronic, decays. However, in certain models (e.g.,
the SUSY extensions discussed in Ref. [131], or in the MLRS), the muon and neutron de-
cay derived limits become comparable [142]. In order to illustrate the relative sensitivities
of the muon and neutron sectors, we have attempted to translate the muon limits from
Refs. [10] and [141] into corresponding neutron observables such as LS/LV, LT/LA, and
RT/RA. In doing so we neglected possible differences in SUSY contributions to muon
and quark decays, making the comparison merely illustrative. These limits are plotted in
Figs. 2.7 to 2.10, as appropriate, showing that neutron decay experiments at their current
and projected future sensitivity are not only complementary, but also competitive to the
muon sector.
Limits similar to the ones discussed in Sec. 2.3.3 can be extracted from pion decays
(added complexity of heavier meson decays limits their sensitivity). The presence of a
tensor interaction would manifest itself both in the Fierz interference term in beta decays
(e.g., of the neutron) and in a non-zero value of the tensor form factor, FT, for the pion.
The latter was hinted at for well over a decade, but was recently found to be constrained
to −5.2 × 10−4 < F < 4.0 × 10−4T with 90% C.L. [143]. While values for b in neutron
beta decay and for the pion form factor FT are not directly comparable, in certain simple
scenarios they would be of the same order [144]. Thus, finding a non-zero value for b in
neutron beta decay at the level of O(10−3) would be extremely interesting. Similarly, the
π → e + ν decay (πe2) offers a very sensitive means to study non-(V−A) weak couplings,
primarily through a pseudoscalar term in the amplitude. Alternatively, the πe2 decay
provides the most sensitive test of lepton universality. Thus, new measurements in neutron
decay would complement the results of precision experiments in the pion sector, such as
PIBETA [145] and PEN [146].
Right-Handed Coupling Constraints from
Neutrinoless Double Beta Decay and Neutrino Mass
The most natural mechanism of neutrinoless double beta decay (ββ0ν) is through virtual
electron-neutrino exchange between the two neutron decay vertices. The LH and RH νe
may mix with mass eigenstate Majorana neutrinos Ni [147]:
∑6 1− 6γ ∑ 1 + γ
νeL =
5 5
Uei Ni and ν2 eR
= Vei Ni, (2.127)2
i=1 i=1
where Uei and Vei denote elements of the LH and RH mixing matrices, respectively.
The ββ0ν decay amplitude with the virtual neutrino propagator has two parts [148].
If the SM LH V−A coupling combines with LH coupling terms (LL interference), the
amplitude contribution is proportional to the Majorana neutrino masses (weighted with
the U2ei factors). Since from neutrino oscillations we have rather small lower limits for
these masses (40meV for the heaviest LH neutrino [149]), we get only weak constraints for
the non-SM LH couplings. On the other hand, if the SM LH V−A coupling combines with
RH non-SM terms (LR interference), the amplitude is proportional to the virtual neutrino
momentum (instead of the neutrino mass); and since the momentum can be quite large we
2.4. PREVIOUS AND COMPETING MEASUREMENTS OF A 35
get constraints for the RH non-SM couplings. The latter part of the ββ0ν decay amplitude
is proporti∑onal to the effective RH couplings R̃j = Rjε, for j =V,A,S,T, where [147, 150]6
ε = (light)UeiVei. (2.128)
i=1
Here, “light” implies that the sum is over the light-mass neutrinos with mi < 10MeV only.
According to Ref. [147] there are three different scenarios:
D: all neutrinos are light Dirac particles
=⇒ no constraints for non-SM couplings, because ε = 0.
M-I: all neutrinos are light (< 1MeV) Majorana particles
=⇒ no constraints for non-SM couplings, because ε = 0 from orthogonality condi-
tion.
M-II: both light (<MeV) and heavy (>GeV) Majorana neutrinos exist
=⇒ constraints for non-SM couplings: ε 6= 0, because heavy neutrinos are missing
from the sum; ε is on the order of the unknown, likely small, mixing angle θLR
between LH and RH neutrinos.
In the M-II scenario there are stringent constraints for the effective RH V,A,S,T cou-
plings: |R̃j | < 10−8 [148]. These effective couplings are proportional to ε ∼ θLR [147, 150].
Since ε depends on specific neutrino mixing models, it is not possible to give model inde-
pendent limits for the Rj couplings based on ββ0ν decay data. We have already mentioned
in Sec. 2.3.1 that for the heavy RH (Majorana) neutrinos the RH observables in neutron
decay are kinematically weakened or for special cases completely suppressed.
Assuming 1TeV effective RH neutrino mass scale within M-II, one obtains |ζ| < 4.7×
10−3 and m2 > 1.1TeV [150]. For a larger RH neutrino mass scale these constraints
become weaker.
In Ref. [151] Klapdor-Kleingrothaus et al. argued that ββ0ν decay occurs in nature.
If further experiments confirm this observation, one can be sure that the neutrinos are
Majorana particles.
The RH couplings can contribute to neutrino mass through loop effects, leading to
constraints on the RH coupling constants from neutrino mass limits [152]. Using the
absolute neutrino mass limit m(νe) < 2.2 eV from the Troitsk and Mainz tritium decay
experiments [153, 154], one obtains the 1σ limits:
|RS| < 0.01, |RT| < 0.1, and |RV −RA| < 0.1. (2.129)
With the m(νe) < 0.22 eV model dependent limit from cosmology36 [156], the above
coupling constant limits become 10 times more restrictive. An intermediate neutrino
mass upper limit in the order of (0.5 − 0.6) eV comes from ββ0ν decay [151] and from
other cosmology analysis [157].
2.4 Previous and Competing Measurements of a
2.4.1 Previous Measurements
Previous experiments have obtained consistent measurements of the neutrino-electron cor-
relation coefficient a with 5% accuracy. As discussed earlier in Chap. 1 and Sec. 2.1.5, the
36The sensitivity of the KATRIN experiment [155] is similar.
36 CHAPTER 2. NEUTRON BETA DECAY
Figure 2.11: The apparatus of Grigor’ev et al.: 1-double toroidal spectrometer, 2-ellipsoidal pro-
ton mirror, 3-electron multiplier (proton detector), 4-photomultiplier (electron de-
tector), 5-Geiger counter, 6-double coincidence trigger circuit, 7-camera control, 8-
oscilloscope, 9-boron carbide neutron absorber, 10-lead shield. Figure from Ref. [158].
determination of a is more difficult than measurements of the beta asymmetry parameter
A, since the neutrino is hard to detect and low-energy (T < 752 eV) recoil protons have
to be detected instead.
The Measurement of Grigor’ev et al. at the ITER Research Reactor
The first important measurement of a was carried out at the ITER research reactor in
the Soviet Union in 1967 by Grigor’ev et al. [158]. A scheme of the apparatus is shown
in Fig. 2.11. The proton recoil spectrum was measured in coincidence at a fixed electron
energy, in order to reduce backgrounds. The proton energy was measured by TOF through
a focusing ellipsoidal electrostatic mirror. Electron and proton events coincident within a
5.5µs time window triggered a camera that photographed the oscilloscope traces of both
signals. The result was
a = −0.091(39), (2.130)
with a relative error of 43% dominated by counting statistics. At that time, the experiment
of Grigor’ev et al. gave the best determination of |λ| = 1.22(8).37
The Measurement of Stratowa et al. at the Research Center Seibersdorf
The first precision measurement of the parameter a was conducted at the ASTRA reactor
in Vienna by Stratowa et al. [40]. The set-up is schematically shown in Fig. 2.12. Stratowa
et al. measured a from the shape of the proton recoil spectrum. The neutron decay volume
was in-pile near the core of the reactor. A throughgoing beam tube was used, with no direct
view to the reactor core or onto the moderator, to reduce background. Decay protons,
37According to Eq. (2.25), a can determine the magnitude but not the sign of λ.
2.4. PREVIOUS AND COMPETING MEASUREMENTS OF A 37
Figure 2.12: The experiment of Stratowa et al.. The neutron decay volume was in-pile near the
core of the reactor. Decay protons emerging from the beam hole were analyzed in a
spherical electrostatic spectrometer. Figure taken from Ref. [40].
Figure 2.13: The experiment of Byrne et al.. A cold neutron beam passed through the center of a
quasi-Penning trap. The decay protons emerged from a high to a low magnetic field
region. Their energy was measured by means of a superimposed electric field. Figure
taken from Ref. [41].
accepted only when their momentum was almost parallel to the tube, were analyzed in
a spherical electrostatic spectrometer. The experiment achieved a proton count rate of
89 s−1, and the result was
a = −0.1017(51), (2.131)
with a 5% relative error dominated by systematic effects, especially the proton detector
energy calibration, corrections for the thermal neutron motion, and proton scattering from
residual gas. A result that was not surpassed for two decades.
The Measurement of Byrne et al. at the Institut Laue-Langevin
The second and latest precision measurement of a was performed at the Institut Laue-
Langevin (ILL) in Grenoble by Byrne et al. [41, 159], with an apparatus converted from
a previous neutron lifetime experiment [160]. A sketch of the experiment is shown in
Fig. 2.13. The angular correlation coefficient a was extracted from the integral proton
spectrum (for details see the measurement principles of the aSPECT spectrometer in
Sec. 3.1). A cold38 neutron beam passed through the center of a quasi-Penning trap. The
decay protons emerged from a high to a low magnetic field region. In this way, their
38Free neutrons are classified according to their kinetic energy. Cold neutrons have an energy from
0.5µeV to 0.025 eV, whereas, e.g., fast neutrons from fusion reactions have an energy > 1MeV.
38 CHAPTER 2. NEUTRON BETA DECAY
Figure 2.14: A sketch of the electromagnetic set-up of the aSPECT experiment at the FRM II. The
magnet coils (c1−c11) surround the main vacuum system. The electrodes (e1−e17)
are installed inside the vacuum system. The analyzing plane is a local magnetic field
maximum; it is not made of any material. We note that the figure is rotated by 90 °.
For details see the following chapter. Figure taken from Ref. [33].
transversal momentum was transformed into longitudinal momentum. The proton energy
was then measured by means of a superimposed electric field. The result was
a = −0.1054(55), (2.132)
with a 5% relative error, similar to Stratowa et al. both in value and precision. Some
of the weaknesses of the experiment were incomplete energy transfer from transverse to
longitudinal motion, and violation of the adiabatic conditions.
First Measurements with aSPECT at the FRM II
First measurements of the parameter a with the aSPECT spectrometer were carried out
at the particle physics beam MEPHISTO at the Forschungsneutronenquelle Heinz Maier-
Leibnitz (FRM II) in Munich by Baeßler et al. [33]. The set-up is schematically shown in
Fig. 2.14. The aSPECT experiment [46, 47] is a proton spectrometer, similar to Byrne
et al. in approach but designed to correct some of its weaknesses. In short, aSPECT
is a retardation spectrometer which measures the proton recoil spectrum by counting
decay protons that overcome an electrostatic barrier, UA. For details on the aSPECT
spectrometer see the following chapters.
In our first beam time during 2005/2006 we studied the properties of the spectrometer
[33, 49]. The most serious problem turned out to be situation- and time-dependent
behavior of the background, cf. Fig. 2.15. For data sets, in which an unexpected
proton-like peak is visible in pulse height spectra with UA = 800V, the extracted value of
a is shifted to more negative values, as can be seen from Fig. 2.15b39. From those data
sets in which a background problem was not obvious, we could extract a value of a, but
we could not quantify the background correction nor its uncertainty (see also page 56
under “Penning Traps and Penning Discharge” and Sec. 5.3).
To sum up, the present best experiments have an uncertainty of ∆a/a = 5% and since
the late 1970s there is no substantial improvement. The measurements of Grigor’ev et al.,
39Please note that Fig. 2.15b slightly differs from Fig. (4.6) in Ref. [49], as these are not the final results
of the data analysis.
2.4. PREVIOUS AND COMPETING MEASUREMENTS OF A 39
(a) (b)
Figure 2.15: (a) Pulse height spectra measured at the FRM II with a silicon PIN diode set at
a potential of -30 kV. With increasing barrier voltage the count rate in the proton
peak (right peak) decreases whereas the electronic noise (left peak) is not influenced.
The background is measured at UA = 800V. (b) Extracted value of the angular
correlation coefficient a versus background count rate. For data sets, in which an
unexpected proton-like peak is visible in pulse height spectra with UA = 800V, as
shown in (a), the extracted value of a is shifted to more negative values. Error bars
show statistical errors only. For comparison, the gray bar represents the Particle
Data Group’s (PDG) 2011 average [10].
Stratowa et al., and Byrne et al. yield a world average of [10]:
a = −0.103(4). (2.133)
Here, the Particle Data Group does not use the data from [161] for their average as
Mostovoi calculates a from its measurement of λ = gA/gV. Figure 2.16 shows the weighted
average from the PDG 2011 review.
2.4.2 Upcoming Experiments
As discussed earlier in Chap. 1 there is strong motivation to improve the uncertainty
in the neutrino-electron correlation coefficient a to less than 1%. Two major, funded
experiments, based in the United States are currently attempting to do this: aCORN
[98, 162] at the National Institute of Standards and Technology (NIST) and Nab [99, 163]
at the new Spallation Neutron Source (SNS) in Oak Ridge, Tennesse.
The aCORN Experiment at NIST
The aCORN experiment [98, 162] uses a new method first proposed by Yerozolimsky and
Mostovoy [164, 165]. A sketch of the spectrometer is shown in Fig. 2.17a. Decay electrons
and protons are detected in coincidence. The electron energy and the TOF between
electron and proton detection will be measured, to discriminate between two groups of
protons, as shown in the top of Fig. 2.17b. Group II (protons first emitted in the same
direction as the electrons towards the electron detector) will be slower than the other group
I (protons emitted in direction of the proton detector opposite the electron detector). The
method relies on the construction of an asymmetry that directly yields a without
40 CHAPTER 2. NEUTRON BETA DECAY
Figure 2.16: World average of the neutrino-electron correlation coefficient a from the PDG 2011
review [10].
(a) (b)
Figure 2.17: (a) A sketch of the aCORN apparatus. (b) Measurement principle of aCORN: MC
simulation of the proton TOF versus the electron energy. The two groups I (lower
branch) and II (upper branch) are well separated for electron energies below 300 keV.
Figures taken from Refs. [22, 98]
2.4. PREVIOUS AND COMPETING MEASUREMENTS OF A 41
(a) (b)
Figure 2.18: (a) A sketch of the set-up of an asymmetric Nab spectrometer [163]. (b) Measurement
principle of Nab: Proton phase space, in terms of p2p, as a function of electron kinetic
energy. Figures taken from Refs. [99, 163]
requiring precise proton spectroscopy:
1 N −N
a(Ee) = K(Ee)
I II
, (2.134)
ve NI +NII
where K(Ee) is a calculated instrumental constant, and NI and NII denote the number of
events in group I and group II, respectively.
The apparatus was integrated and tested at the Indiana University Cyclotron Facility
(IUCF) and then moved to the NIST Center for Research for the initial run. All systems
work, but problems of high voltage discharges, due to crossed electric and magnetic fields,
require a redesign of the proton focusing for low magnetic field [166]. The experiment
hopes to obtain a relative uncertainty in a of 1%.
The Nab Experiment at the SNS
The Nab experiment [99] is based on an observation made by Bowman [167]. The new
asymmetric version of the instrument (for details see [163]) is conceptually shown in
Fig. 2.18a. Decay electrons and protons are detected in coincidence. Similar to aCORN,
the electron energy and the time between electron and proton detection, tp, will be mea-
sured. The Nab method relies on the linear dependence of cos θeν on p2p for a given electron
momentum p 40e (or kinetic energy Ee) :
p2p = p
2
e + 2p
2
epν cos θeν + pν . (2.135)
Since pν depends almost only on Ee (or pe), Eq. (2.135) reduces to a linear relation between
cos θ and p2eν p for a fixed pe; the mapping is graphically shown in the bottom of Fig. 2.18b.
Hence, the parameter a is determined from the slopes of the 1/t2p distributions for different
values of Ee.
40Please note that Ee denotes, here, the electron kinetic energy.
42 CHAPTER 2. NEUTRON BETA DECAY
The Nab Collaboration develops a novel electromagnetic spectrometer. The goal is to
measure a and the Fierz interference term b in the same apparatus, with ∆a/a ≈ 10−3
and ∆b ≈ 3× 10−3.
In summary, the measurement of the neutrino-electron correlation coefficient a provides
one of best avenues for an independent determination of λ. The prospects for a significantly
improved value for a in the near future are promising.
Chapter 3
The aSPECT Experiment
The neutron decay spectrometer aSPECT [46, 47] has been designed to perform precise
measurements of the neutrino-electron correlation coefficient a, by measuring the proton
recoil spectrum in the decay of free, unpolarized neutrons. aSPECT is a retardation
spectrometer which, in turn, measures the proton recoil spectrum by counting all decay
protons that overcome an electrostatic barrier, UA, similar to the experiment of Byrne et
al. [41].
After a short introduction to the measurement principles, the electromagnetic set-up
of the aSPECT spectrometer and its detection system will be presented in this chapter.
The main emphasis of this thesis lies on the investigation of several different systematic
effects. Hence, we also introduce the dominant uncertainties in the determination of a.
More details on the electromagnetic set-up and the detection system are found in the
theses of F. Ayala Guardia [53, 168] and M. Simson [52], respectively.
3.1 Measurement Principles
The aSPECT experiment is schematically shown in Fig. 3.1. Unpolarized, cold neutrons
are coming from the left and pass through the decay volume (DV) where about 10−8 of the
neutrons decay. The decay protons are guided by the strong magnetic field towards the
analyzing plane (AP) and subsequently the proton detector on top of the spectrometer.
Protons emitted in the negative z-direction are reflected by the electrostatic mirror on
the bottom of the spectrometer, ensuring 100% acceptance for decay protons. Therefore,
the electrostatic mirror is held at a positive voltage UM > Tp,max. In the AP, situated in
the center of a 54 cm long cylindrical electrode, we apply a variable barrier voltage UA.
Only protons with sufficient kinetic energy can pass the potential barrier. These protons
are accelerated by a high voltage of −15 kV and magnetically focused onto the detector
where they are counted. The detector has to be held at a high negative potential to
post-accelerate the protons to detectable energies and to ensure that they overcome the
magnetic mirror (discussed in the following Sec. 3.1.1) right in front of the detector. The
AP voltage UA is varied between 0 and +800V, to scan the proton spectrum shape, cf.
Fig. 2.3. Measurements with UA = 780V serve to quantify the background. The magnetic
field in the AP, BA, is much lower than in the DV, B0. Therefore, the momenta of the
decay protons are aligned by the inverse magnetic mirror effect. Then the response of
the spectrometer can be described by the so-called transmission function, introduced in
Sec. 3.1.2. In between the DV and the AP a dipole electrode creates a transverse electric
43
44 CHAPTER 3. THE ASPECT EXPERIMENT
Figure 3.1: Scheme of the aSPECT experiment: Unpolarized, cold neutrons (green) pass through
the decay volume (pink) where only a few neutrons decay. The decay protons (red)
are guided by the strong magnetic field towards the analyzing plane (gold) and subse-
quently the proton detector (dark gray). The analyzing plane voltage UA is varied be-
tween 0 and +800V, to scan the proton spectrum shape, cf. Fig. 2.3. Protons emitted
in the negative z-direction are reflected by the electrostatic mirror (black). Trapped
protons are removed from the flux tube with the lower dipole electrode (turquoise).
The upper dipole electrode (purple) serves to align the neutron beam on the detector.
field and hence a large E×B. This is used to sweep out decay protons that would otherwise
be trapped. In between the AP and the proton detector a second dipole electrode, rotated
90° in the vertical relative to the first one, serves to align the neutron beam on the detector.
3.1.1 Adiabatic Invariance and Magnetic Mirror Effect
Adiabatic Invariance
Adiabatic invariance, an approximate conservation law, is a general result for any dynam-
ical system that can be described by a Hamiltonian H(q,p, λ) and which follows periodic
3.1. MEASUREMENT PRINCIPLES 45
motion. Here, q and p denote the generalized coordinates and their canonically conjugate
momenta, respectively, and λ = λ(t) are the system’s parameters. If λ is changed adia-
batically, i.e., the timescale of changes in λ is much greater than the oscillation period,
then the action integrals Ji ∮
Ji(E, λ) = Ji(E(t), λ(t)) = pidqi, i = 1, 2, 3, (3.1)
are the adiabatic invariants of the system, where the contour integrals are taken at fixed
total energy E and fixed value of λ.
Charged Particle Motion
In the aSPECT experiment, the decay products, electrons and protons, are moving in a
static magnetic field, B. The Lorentz force1
FL = q (v ×B) (3.2)
therefore forces the charged particles to move on circular orbits, i.e., to gyrate around
a magnetic field line. Here, q and v are the charge and the velocity of the particle,
respectively. The radius of gyration, rg, is determined by equating the magnitude of the
Lorentz force with the centripetal force
mv
= ⊥rg , (3.3)
eB
where the velocity component v⊥ perpendicular to the magnetic field is given by
v2 = v2|| + v
2
⊥. (3.4)
Here, m is the particle’s mass and v|| is the velocity component parallel to the magnetic
field. The maximum radius of gyration is achieved for particles emitted perpendicular to
the magnetic field:
√
mvmax 2mT
rg,max = =
max
, (3.5)
eB eB
where vmax and Tmax are the maximum velocity and kinetic energy of the particle, re-
spectively. Assuming B = B0 = 2.2T as in Fig. 3.1, the maximum radii of gyration
are
rg,max ≈ 1.80mm (3.6)
both for protons and electrons.
Adiabatic Invariants
Let us return to the adiabatic invariants. In the case of particle motion in an inhomo-
geneous magnetic field, the condition for adiabatic transport can be formulated as the
requirement that th∣∣e qu∣∣antity , defined in [169] as:2πp cos θ ∂B
 = ∣∣ ∣∣ , (3.7)eB2 ∂z
1Please note that we assume w.l.o.g. c = 1; see also our note on page 16 in Chap. 2.
46 CHAPTER 3. THE ASPECT EXPERIMENT
is small (  1). Here p is the particle’s momentum, θ its angle to the z-axis, and ∂B∂z a
small magnetic field gradient in z-direction. According to Ref. [170], then the adiabatic
invariants are
Br2g , (3.8)
p2⊥ , and (3.9)
B
γµ , (3.10)
where p⊥ is the momentum component perpendicular to the magnetic field, γ = √ 1 is
1−β2
the relativistic factor, with the particle’s velocity in units of the velocity of light, β = vc ,
and
mv2 2⊥ eωBrgµ = = (3.11)
2γB 2
is the orbital magnetic moment, with the Larmor frequency ωB = eBγm . By substituting
Eq. (3.8) into Eq. (3.5) we get the maximum radius of gyration for any other strength B
of the magnetic field √
2.2T
rg,max(B) = rg,max(B = 2.2T) · . (3.12)
B
Magnetic Mirror Effect
Similarly, by substituting Eq. (3.9) into Eq. (3.4) we get
v2
v2 2 2||(z) = v − v⊥(z) =
2 − ⊥,0v ·B(z), (3.13)
B0
where v⊥,0 and B0 are the transverse velocity component and the strength of the mag-
netic field, respectively, at, e.g., the decay point P0, and v||(z), v⊥(z), and B(z) are the
longitudinal and transverse velocity components and the strength of the magnetic field,
respectively, at any other point P = (x, y, z) of the particle’s trajectory. Here, we assume
that the dependence of both velocity and magnetic field on the coordinates x and y is
negligible. Two cases must be considered:
Magnetic mirror effect: If a particle is moving towards an increasing magnetic field
B(z), its velocity component v||(z) parallel to the magnetic field decreases, becomes
zero at a certain point, and finally changes its sign. The latter is equivalent to the
reflection at a magnetic mirror. This effect can be used in magnetic bottles, where
charged particles are trapped between two magnetic barriers.
Inverse magnetic mirror effect or magnetic adiabatic collimation: On the other
hand, if a particle is moving towards a decreasing magnetic field B(z), its velocity
component v||(z) parallel to the magnetic field increases. This is equivalent to the
alignment of the particle’s momentum with the magnetic field.
In aSPECT, the decay protons are aligned in the AP by the inverse magnetic mirror
effect. The electrostatic barrier UA is only sensitive to the longitudinal momentum of the
protons. Therefore, the inverse magnetic mirror effect serves to transfer almost the total
proton momentum into longitudinal momentum without loss of momentum.
3.1. MEASUREMENT PRINCIPLES 47
We note that the principle of magnetic adiabatic collimation was first applied in elec-
tron spectroscopy [169, 171, 172]. Starting in the mid 1980 s, this principle was successfully
employed at Troitsk and Mainz in direct neutrino mass measurements via the endpoint
of the tritium beta spectrum [153, 154]. The upgrade of the neutrino mass spectrometer,
KATRIN [155], is based on the same principle [173, 174]. Currently, the WITCH exper-
iment searches for exotic interactions by investigating the neutrino-electron correlation
coefficient a in several allowed nuclear beta decays. For the measurement of the recoil en-
ergy spectrum, WITCH uses an electromagnetic retardation spectrometer with magnetic
adiabatic collimation. As a first step, the recoil ions from the β−-decay of 124In, stored in
a Penning trap, have been investigated [175].
3.1.2 The Adiabatic Transmission Function
The aSPECT spectrometer has been designed such that all decay protons fulfill the con-
dition for adiabatic transport. Therefore, the relative spatial changes of the magnetic and
also electric field have to stay small during one particle gyration [174] (see also [176, 177]):
∆B/B  1 and ∆E/E  1. (3.14)
Then Eqs. (3.8) to (3.10) are still the adiabatic invariants of the system.
Similar to the proton’s velocity, it is useful to decompose the proton kinetic energy Tp
into longitudinal and transverse components, at any point P of the proton’s trajectory:
Tp,||(z) = T (z) cos
2
p θ(z) and (3.15)
Tp,⊥(z) = T (z) sin2p θ(z). (3.16)
Then we can rewrite the orbital magnetic moment Eq. (3.11) in terms of the proton kinetic
energy2:
mv2p,⊥(z) T 2( ) = = p,⊥
(z) T (z) sin θ(z)
µ z = p . (3.17)
2B(z) B(z) B(z)
By substituting Eq. (3.10) into Eq. (3.17) we get the orbital magnetic moment at any
point P of the proton’s trajectory:
T0 sin2 θ0 Tp(z) sin2 θ(z)
µ = = , (3.18)
B0 B(z)
where T0 and θ0 are the proton kinetic energy and its polar angle, respectively, at the
decay point P 20. Solving Eq. (3.18) for sin θ(z) yields:
2 B(z) Tsin ( ) = 0θ z sin2 θ0, (3.19)
B0 Tp(z)
and subsequently for the longitudinal component Eq. (3.15) of the proton kinetic energy:
( )
T ad
B(z)
|| (z) = Tp(z) cos
2 θ(z) = Tp(z) 1− sin2 θ(z) = Tp(z)− T0 sin2 θ0. (3.20)
B0
2Here and in the following, we assume w.l.o.g. γ = 1 because of the low energies of the protons.
48 CHAPTER 3. THE ASPECT EXPERIMENT
Here, the superscript ad denotes “in adiabatic approximation”. For reasons of energy
conservation
Ep = Tp(z) + Vp(z) = T0 = Tp(z) + e (U(z)− U0) , (3.21)
where Vp is the potential energy of the proton, we can deduce the proton kinetic energy
at any point P:
Tp(z) = T0 − e (U(z)− U0) . (3.22)
Substituting this into Eq. (3.20) yields:
B(z)
T ad|| (z) = T0 − e (U(z)− U0)− T0 sin
2 θ0. (3.23)
B0
A decay proton reaches the proton detector if and only if the adiabatic transmission
condition
T ad
!
|| (z) > 0 (3.24)
is fulfilled for every point P along its trajectory. The other way round, if at some point
T ad|| (z) < 0 then the proton will be reflected at this point. Therefore, an important
design goal of aSPECT was that no proton with sufficient kinetic energy to overcome the
electrostatic barrier can be reflected outside the AP, except for the electrostatic mirror.
Let us confine ourselves to the AP. Solving inequation (3.24) for T0, where T ad|| (z) is
given by Eq. (3.23), yields: ( )
B −1A
T0 > Ttr(θ0) = e (UA − U0) 1− sin2 θ0 . (3.25)
B0
The adiabatic transmission energy, Ttr(θ0), is sho(wn in Fig). 3.2. Hence, protons with−1
T > Tmax
BA
0 tr = Ttr(θ0 = ±90°) = e (UA − U0) 1− (3.26)B0
are transmitted through the potential barrier UA, whereas protons with
T < Tmin0 tr = Ttr(θ0 = 0°) = e (UA − U0) (3.27)
are reflected at the potential barrier and subsequently trapped between the AP and the
electrostatic mirror, both independent of their (initial) polar angle. Solving inequation
(3.24) for θ0, in turn, yields that, for initial energies Tmin < T < Tmaxtr 0 tr , protons can pass
the potential barrier only√if the(ir (initial) polar a)ngle goes below
3
max B0 − e (U − U )θ0 < θtr = arcsin 1
A 0
, (3.28)
BA T0
3Please note that, here, we need to consider only polar angles θ ∈ [0, 90°] as protons emitted in the
negative z-direction, i.e., with angles θ ∈ (90°, 180°], are reflected at the electrostatic mirror into angles
(180°− θ) ∈ [0, 90°).
3.1. MEASUREMENT PRINCIPLES 49
Figure 3.2: The adiabatic transmission energy Eq. (3.25) separates the reflection region (red) from
the transmission region (white). In the adiabatic approximation, decay protons with
an initial kinetic energy T0 > Ttr(θ0) will be transmitted by the potential barrier UA,
whereas those with T0 < Ttr(θ0) will be reflected. The transmission line Ttr(θ0) is
shown for UA − U0 = 400V and BA/B0 = 0.203. We mention that, if the adiabatic
transmission condition Eq. (3.24) is fulfilled for θ0 = 90°, it is fulfilled also for any
other (initial) polar angle θ0.
i.e., protons with an initial energy of Tmintr < T0 < Tmaxtr are transmitted through the
potential barrier wi∫th a probability of4θmax ( √ )tr [ ]θmax
− trwtr(T0) = √dθ sin θ (= cos θ =) 1− 1− sin
2 θmaxtr
0 0
B e (U − U )
= 1− 1− 0 1− A 0 . (3.29)
BA T0
Altogether, the probability that a proton with given definite kinetic energy T0 can pass
the AP is described bythe transmission function: 0 √ ( ) , T ≤ Tmin0 tr
Ftr(T0;U ) =  1− 1− B0 1− e(UA−U0)A  B , Tmin < T0 ≤ Tmax .A T0 tr tr1 , T > Tmax0 tr
(3.30)
In the ideal case, BA/B0 −→ 0, the transmission function becomes a step function (see
also Eqs. (3.26) and (3.27)). We have chosen BA/B0 = 0.203 as a reasonable compromise
between size and resolution of the spectrometer. In this case, deviations from the adiabatic
approximation can be neglected [47]. The transmission function is shown in Fig. 3.3,
together with the proton recoil spectrum with the recommended value for a = −0.103
[10].
4Please note that the proton recoil spectrum Eq. (2.43) depends neither on the proton’s azimuthal nor
polar angle; the latter because we are investigating unpolarized neutrons. We therefore can derive the
transmission probability by integration over only the polar angle θ.
50 CHAPTER 3. THE ASPECT EXPERIMENT
Figure 3.3: The black and red line indicate the transmission function for protons in unpolarized
neutron decay, for BA/B0 = 0.203 and UA − U0 = 50V respectively UA − U0 =
400V. For elucidation, the green line shows the proton energy spectrum with the
recommended value for a = −0.103 [10]. To demonstrate the sensitivity of the proton
recoil spectrum on the neutrino-electron correlation coefficient a, the blue line shows
the proton spectrum for a different value of a.
We mention that Eq. (3.30) depends only on the electrostatic potential and the mag-
netic field values in the DV and the AP. More precisely, in the adiabatic approximation,
the transmission function depends only on the ratio of the magnetic fields in the AP and
the DV,
rB = BA/B0, (3.31)
and the barrier voltage UA; the latter as in our measurement the DV is grounded, i.e.,
U0 ≡ 0. Only these two quantities have to be determined with high precision [46]. At all
other points of the proton’s trajectory, a rough knowledge of the electrostatic potential
and the magnetic field is sufficient, provided that the adiabatic transmission condition
Eq. (3.24) is fulfilled everywhere. Furthermore, the transmission function does not depend
on the detector characteristics, as long as the detector just counts the decay protons which
pass the AP.
3.1.3 Computation of a from the Proton Recoil Spectrum
In our experiment, the proton detector counts all decay protons that overcome the elec-
trostatic barrier. The proton count rate at a given barrier voltage UA can then be derived
from Eq. (3.30): ∫ Tp,max
Np(UA) = N0 dTFtr (T ;UA)wp,C,α(T ), (3.32)
0
where N0 = Np(UA = 0V) is the full proton decay rate. Figure 3.4 shows the proton count
rate Np(UA) for two different values of a to demonstrate the sensitivity of the integral
3.1. MEASUREMENT PRINCIPLES 51
Figure 3.4: The theoretical integral proton spectrum Eq. (3.32) in neutron beta decay, for
BA/B0 = 0.203. The green line is the prediction from the SM with the recommended
value for a = −0.103 [10], the blue line shows how a deviation from that (a = +0.3)
would look like.
proton spectrum on the angular correlation coefficient a. We measure the proton count
rate Np(UA) at different barrier voltages UA and extract the neutrino-electron correlation
coefficient a from a two parameter fit to Np(UA). The two fit parameters are N0 and of
course the angular correlation coefficient a. This is our standard method to determine a
from the measured proton spectra.
In Ref. [46], the specific count rate ratio
N
( ) = p
(UA)
rh UA (3.33)
N0
was considered and its sensitivity to a was demonstrated. If in Eq. (3.32) we now replace
the proton recoil spectrum wp,C,α(T ) by Nachtmann’s formula Eq. (2.38), wp,[Nac68], we
obtain for the co∫unt rate ratioT ∫ ∫p,max T0 dTFtr(T ;UA)g1(T ) + a ∫ p,max0 dTFtr(T ;UA)g2(T )rh(UA) = . (3.34)Tp,max T
0 dTg1(T ) + a
p,max
0 dTg2(T )
Solving Eq∫. (3.34) for a yields:∫T ∫p,max0 dTFtr(T ;UA)g1(T )− rh(U ) ∫Tp,maxA 0 dT g= 1(T )a . (3.35)Tp,max
0 dTFtr(T ; ) ( )− ( )
T
U g T r U p,maxA 2 h A 0 dTg2(T )
Thus, the angular correlation coefficient a can also be determined from measurements of
the decay rate ratio rh(UA) for different barrier voltages UA. Then,
an absolute uncertainty of the potential barrier UA of 7mV and a
relative uncertainty of the magnetic field ratio rB of 1 × 10−4 would
each correspond to a relative uncertainty in a of about 0.1%,
52 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.5: Relative change of the angular correlation coefficient a for (a) a relative uncertainty of
the magnetic field ratio rB of 1×10−4 and (b) an absolute uncertainty of the potential
barrier UA of 10mV. Each value corresponds to a relative uncertainty in a of about
0.1%. We mention that the uncertainty in a strongly depends on the barrier voltage
chosen. Input data for the simulation: BA/B0 = 0.203 and a = −0.103 [10].
(a) Full view (b) Zoom
Figure 3.6: Relative change of the angular correlation coefficient a for (a) different uncer-
tainties of the potential barrier UA, for a special set of barrier voltages UA =
50, 250, 400, 500, 600V. The (b) zoom to uncertainties in the order of 10mV shows
that an absolute uncertainty of the potential barrier UA of 10mV would correspond
to a shift in a of ∆a/a = +0.12(5)%. Input data for the simulation: BA/B0 = 0.203
and a = −0.103 [10]. The error bars, in the order of 0.05%, represent the error by
fitting Eq. (3.32) to the simulated proton count rates only.
as can be seen from Fig. 3.5 (see also [46]). As one can also see in Fig. 3.5, the uncertainty
in a strongly depends on the barrier voltage chosen. For measurements of the integral
proton spectrum Np(UA), we therefore have to choose a suitable set of barrier voltages.
An example for a special set of barrier voltages, UA = 50, 250, 400, 500, 600V, is presented
in Fig. 3.6. The figure shows the relative change in a for different uncertainties of the
potential barrier.
3.2. THE RETARDATION SPECTROMETER ASPECT 53
Background, caused by positive ions coming from the decay region, may be efficiently
reduced if measurements of the proton count rates Np(UA) are taken for UA not less than
≈ 40V (for details see Ref. [47] and also Secs. 3.4.3 and 4.2.4). Therefore, the decay
rate ratio rh(UA) must be replaced by a decay rate ratio r50 Vh (UA) relative to the proton
count rate Np(UA = 50V). Analogous to Eq. (3.35), we then get a as a function of the
transmissio∫n function:∫ T ∫p,max dTF (T ;U )g (T )− r50 V( ) ∫ TU p,max= 0 tr A 1 h A 0 dTFtr(T ; 50 V)g1(T )a . (3.36)Tp,max dTF (T ;U )g (T )− r50 V0 tr A 2 h (UA) Tp,max0 dTFtr(T ; 50 V)g2(T )
In contrast to our standard method, the last two methods are only used to study several
different systematic effects.
In reality, deviations from adiabatic transport, proton scattering on residual gas
molecules, and many other systematic effects may influence the transmission function
and hence falsify the interpretation of the measured proton spectra. These issues will be
discussed in Sec. 3.4.
3.2 The Retardation Spectrometer aSPECT
In fact, the electromagnetic set-up of the aSPECT spectrometer is much more complex
than shown previously in Fig.3.1. A detailed sketch of the set-up is shown in Fig. 3.7. The
high number of field shaping coils and electrodes follows from the requirement Eq. (3.14)
that changes in the magnetic and electric fields are kept small. In this way, the adiabatic
approximation can be used to calculate the proton’s trajectories [47].
3.2.1 The Electrode System
The aSPECT spectrometer consists of a system of eighteen electrodes, denoted as e1
to e17,5 placed inside the cold bore (vacuum) tube of the aSPECT magnet (for details
see the following section). The design of the electrode system is shown in Fig. 3.7. All
electrodes have cylindrical shape, but not all of them are axially symmetric, except for
the DV electrode gr. The aSPECT electrodes were made of oxygen-free high thermal
conductivity (OFHC) copper and electrolytically coated with 2µm of silver6 and then
1µm of gold7, except for the high-voltage electrodes e16 and e17. These two electrodes
were made of stainless steel AISI-No. 316L (EN-norm 1.4404) and electropolished. The
typical voltage settings are listed in Table 3.1. The electrostatic potential along the z-axis,
for these settings, is shown in the bottom of Fig. 3.8.
The Decay Volume
The DV, shown in Fig. 3.9a, consists of three parts, denoted as e3, gr, and e6. All three
parts are connected to ground to shield the DV from electric fields of the other electrodes.
Accordingly the electrostatic potential within the DV is smaller than 1mV, well within
5Please note that we use the (new) naming scheme of Ref. [53].
6The renewed and over-coated electrodes e3 to e6, e12, and e14 were coated with 10µm of silver instead.
7Please note that in Refs. [34, 52] the layer thicknesses are misstated. In addition, we would like to
stress that none of the electrodes was electropolished, except for the high voltage electrodes e16 and e17.
54 CHAPTER 3. THE ASPECT EXPERIMENT
Figure 3.7: A sketch of the electromagnetic set-up of the aSPECT experiment at the ILL. The
magnet coils (blue, c1−c15) surround the main vacuum system. The electrodes (red,
e1−e17) are installed inside the vacuum system. The analyzing plane is a local mag-
netic field maximum; it is not made of any material. For comparison with the magnetic
field and the electrostatic potential values in Fig. 3.8, the figure is rotated by 90°.
Figure 3.8: The values of (top) the magnetic field and (bottom) the electrostatic potential along
the z-axis of the aSPECT spectrometer. The corresponding potentials of the electrodes
e1 to e17 are listed in Table 3.1. Input data for the electromagnetic field calculations:
Imain = 100A, I3 = 50A, I5 = 21.4A, I12 = −I13 = 36.4A, I14 = I15 = 0, U1 = U1b =
800V, U2 = 1000V, U8 = −525V, UA = 400V, U16 = −2 kV, and U17 = −15 kV.
3.2. THE RETARDATION SPECTROMETER ASPECT 55
Table 3.1: Typical voltage settings of the aSPECT electrodes, denoted as e1 to e17.
El.no. Voltage [V] Comment
e1 800 electrostatic mirror (wire system)
e1b 800 electrostatic mirror (holder for wire system)
e2 1000 / 820 electrostatic mirror (quadrupole used as cylinder)
e3 grounded (U0 = 0) DV (bottom cylinder); wired individually (for tests)
gr grounded (U0 = 0) DV electrode; wired individually (for tests)
e6 grounded (U0 = 0) DV (top cylinder); wired individually (for tests)
e7 grounded same ground than DV; usable for systematic tests
e8 -50|-1000 or 0|-200 lower E×B drift, sides R and L (half cylinders)
e9 grounded same than DV ground
e10 0.435331×UA variable
e11 0.683960×UA variable
e12 0.892352×UA variable
e13 0.991040×UA variable
e14 UA AP electrode (variable)
e15 0.985094×UA variable
e15b grounded same ground than spectrometer, not than DV
e16 -2|-2 or -3.7|-4.2×103 upper E×B drift, sides A and B (half cylinders)
e17 -10/-12/-15×103 detector HV
our tolerance of less than 10mV. The neutron beam passes through the rectangular tunnel
gr with a cross-section of 70× 110mm2. The rectangular tunnel, in turn, has an opening
on one side for vacuum pumping8 (to the backside of the photograph). The decay protons
are guided by the strong magnetic field B0 towards the proton detector on top of the
spectrometer (to the top of the photograph) or the mirror on the bottom, through two
cylindrical openings on top and on bottom of the rectangular tunnel. Each of the two
cylindrical openings is followed by a long cylinder, separated from it by a small gap9.
The magnetic field lines connecting the DV and the proton detector define the so-called
flux tube. For elucidation, the blue dashed lines in Fig. 3.7 show calculated field lines.
In comparison with our first beam time at the Forschungs-Neutronenquelle Heinz
Maier-Leibnitz (FRM II) in Munich, Germany, several electrodes were changed in shape.
The main reason was strong discharge phenomena as discussed in the following section.
8Please note that during our latest beam time at the ILL the opening was wrongly installed opposite
to the turbo-molecular pump.
9Please note that in the first half (November/December 2007) of our latest beam time the cylindrical
openings were separated from the long cylinders by means of two ceramics rings, with a smaller inner
diameter than the DV electrodes e3, gr, and e6. These rings were removed just before the main phase
of the data taking (April/Mai 2008), as it was found that in the first half of the beam time the proton
count rates decreased with time. We are convinced that the two ceramics rings got charged by the decay
protons and electrons and that the electric field distribution therefore exhibited a potential barrier above
the neutron beam, comparable to the one shown in Fig. 6.21b.
56 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.9: Photographs of (a) the decay volume (DV) electrodes e3, gr, and e6 and (b) the
electrostatic mirror electrodes e1, e1b, and e2. The photograph (a) also shows two of
the four getter pumps to improve the vacuum conditions (discussed in Sec. 3.4.3) as
well as the mirror electrode e2. See the text for details.
Penning Traps and Penning Discharge
In previous measurements with the aSPECT spectrometer, relatively strong discharge
phenomena have been observed [33, 49] (see also page 38 under “First Measurements with
aSPECT at the FRM II” and Sec. 5.3). These discharges could be due to field emission
processes near the proton detector, where high electric field is present. Furthermore, the
aSPECT electrode system contains several Penning traps [178, 179], both for electrons
and for positive ions, as can be seen from Fig. 3.10. These traps, in turn, could also cause
serious discharge processes. In order to understand these processes, the literature on the
Penning discharge phenomenon was studied and a variety of calculations was performed
to search for various traps, which could be responsible for the discharge and background
processes observed.
In some respect, the Penning discharge is similar to the Townsend discharge, which
occurs for relatively high gas pressure. Although both the vacuum breakdown and the
Penning discharge are present even with very good vacuum conditions, the physics of
the Penning discharge is completely different from that of the vacuum breakdown. The
main difference is that the Penning discharge occurs only in the presence of magnetic
field. It also requires stable charged particle trapping, mainly of electrons. In this process,
trapped electrons can suffer ionization collisions with residual gas molecules even with a
high vacuum in the order of 10−9 mbar. In addition, the positive ions, hitting the anode,
3.2. THE RETARDATION SPECTROMETER ASPECT 57
Figure 3.10: A sketch of the electromagnetic set-up of the aSPECT experiment at the FRM II.
The orange and red regions show possible electron and positive ion traps, respectively,
as discussed in the text. We note that the figure is rotated by 90°. Figure adapted
from Ref. [33].
can produce more electrons by secondary emission.
A detailed comparison of the aSPECT discharge phenomena with those of other exper-
iments, in particular the neutrino mass experiments of Mainz and KATRIN [180–183] (see
also [184, 185]), has been performed and Penning traps in the aSPECT electrode system
have been searched. Three main trapping regions have been found: in the bottom part
of the spectrometer, at the AP, and near the proton detector. At the mirror electrode, a
wire system was installed to prevent electron trapping (for details see the following sec-
tion). At the detector, a different dipole electrode and a different detector electrode were
installed and, simultaneously, the detector potential was decreased, all three to avoid the
deep traps present in previous aSPECT measurements (see page 59 under “The Dipole
Electrodes” for details). The Penning traps at the AP could not be changed, because these
traps are absolutely necessary for the electric retardation method to measure the proton
recoil spectrum. In addition, the traps at the AP are only a few hundred Volts deep.
Further details on the Penning traps and discharges can be found in Refs. [33, 47,
186]. For investigations of fake effects caused by Penning discharges, such as AP voltage
dependent background, see also Ref. [50] and Sec. 5.3.
The Electrostatic Mirror
The electrostatic mirror, shown in Fig. 3.9b, also consists of three parts, denoted as e1,
e1b, and e2. These three parts are typically held at positive potentials
U1 = U1b = 800V and U2 = 1000V, (3.37)
to reflect all decay protons emitted in the negative z-direction.
The quadrupole electrode e2, in turn, consists of four quarter cylinders, denoted as
parts M, N, P, and R (see also Fig. 3.18b), separated from each other by small slits. For
measurements of the neutrino-electron correlation coefficient a, all four parts are powered
by the same power supply; for tests with a calibration source (developed by M. Borg
[34]), which can be flange-mounted to the bottom of the spectrometer, the four parts are
powered individually. This allows us to shift the ion beam leaving the calibration source
in the x-y-plane.
As discussed in the previous section, a wire system (e1) was installed at the mirror
electrode e1b to prevent electron trapping. In order not to interfere with the calibration
58 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.11: Electrostatic potential in the x-y-plane of the wire system e1 for an OFHC wire (a)
with a diameter of 500µm and a wire distance of 10mm and (b) with a diameter of
125µm and a wire distance of 5mm. Main input data for the electric field calculations
(with COMSOL 3.2b Multiphysics): U1 = U1b = 800V and U2 = 1000V. The
comparison between (a) and (b) shows that a wire diameter and distance of 125µm
and 5mm, respectively, ensure a rather homogeneous electric field distribution of at
least 780V over the entire flux tube. Here, the black lines represent the electrodes
e1 to e16, the two heat shields for the detector electrode e17, as well as the vacuum
tube.
source different wire diameters and distances were investigated. A wire diameter of 125µm
and a wire distance of 5mm were chosen. This ensures a rather homogeneous electric field
distribution of at least 780V over the entire flux tube, as can be seen from Fig. 3.11.
The Analyzing Plane
The AP is situated in the center of the electrode e14, as shown in Fig. 3.7. The electrodes
e10 to e15 are powered by the same power supply; the electrode e14 is connected directly
to the power supply, whereas the electrodes e10 to e13 and e15 are connected by means of
a resistor based voltage divider. This ensures that the potential in the center of the AP,
for, e.g., UA = 400V, is only by 2mV smaller than at the surface of the electrode e14 and
hence that the adiabatic transmission condition Eq. (3.24) is always fulfilled (as discussed
in Sec. 3.2.2, cf. Eqs. (3.42) to (3.45)).
The Dipole Electrodes
In our measurement, two dipole electrodes, denoted as e8 and e16, are used. Each of the
two dipole electrodes consists of two half cylinders, denoted as parts R and L respectively
A and B, which may be held at different potentials. A different potential produces an
electric field perpendicular to the magnetic field and this, in turn, a so-called E×B drift.
In addition to their circular motion, the Lorentz force (see also Eq. (3.2) and Fn. 1)
FL = q (E + v ×B) (3.38)
3.2. THE RETARDATION SPECTROMETER ASPECT 59
(a) (b)
Figure 3.12: Photographs of (a) the upper dipole electrode e16, sides A and B, and (b) the detector
electrode e17. The photograph (a) also shows the heat shields to minimize the heat
input from the detector electrode to the cold bore tube of the aSPECT magnet; the
photograph (b) shows also the proton detector (see Sec. 3.3.1 for details).
forces the decay protons10 to a drift perpendicular to both electric and magnetic field,
with a drift velocity of
E×B
vdrift = . (3.39)
B2
The lower dipole electrode e8 is situated between the DV and the AP, as can be seen
from Fig. 3.7. It serves to sweep out all decay protons that cannot pass the potential
barrier and would otherwise be trapped between the electrostatic mirror and the AP.
After several passages through the dipole electrode, these protons will be absorbed by the
surface of an electrode or the cold bore tube, as the direction of drift is independent of the
direction of flight. We mention that, with increasing potential difference between sides R
and L, the trapping time of these protons decreases and consequently the systematic effect
due to proton collisions with the residual gas molecules is reduced (discussed in Sec. 3.4.3).
The upper dipole electrode e16, shown in Fig. 3.12, is situated right in front of the
proton detector (cf. Fig. 3.7). It causes a transverse drift which serves to align the neutron
beam on the proton detector (see also Fig. 3.18b). The dipole electrode has to be held
at a negative potential of at least −600V to ensure that all decay protons that pass the
potential barrier can also overcome the magnetic mirror right in front of the detector.
As discussed earlier on page 56 under “Penning Traps and Penning Discharge”, the two
parallel stainless steel plates (cf. Fig. 3.13) of the dipole electrode’s predecessor were
10Please note that the spatial displacement of charged particles moving in an E ×B drift depends on
the time they spend in the drift. Due to their higher energy but smaller mass the decay electrons pass far
more quickly through an E×B drift. Therefore, the displacement of the decay electrons is negligible.
60 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.13: Electrostatic potential of the upper dipole electrode e16 at the FRM II (a) in the
y-z-plane for x = −1.9 cm and (b) in the x-z-plane for y = ±5 cm. Main input data
for the electric field calculations (with COMSOL 3.2b Multiphysics): U16A = −4 kV,
U16B = 0, and U17 = −30 kV. The gray lines highlight potential Penning traps. We
mention that the traps on the edge of the heat shields (dark gray) are always present,
whereas those close to side B of the upper dipole electrode e16 (light gray) strongly
depend on the voltage applied.
(a) (b)
Figure 3.14: Electrostatic potential of the upper dipole electrode e16 at the ILL (a) in the y-z-plane
for x = 0 and (b) in the x-z-plane for y = 0. Main input data for the electric field
calculations (with COMSOL 3.2b Multiphysics): U16A = −4.2 kV, U16B = −0.2 kV,
and U17 = −15 kV. In the new design of the high voltage electrodes e16 and e17, the
potential Penning traps shown in Fig. 3.13 are no longer present.
3.2. THE RETARDATION SPECTROMETER ASPECT 61
(a) (b)
Figure 3.15: Electric field strength of the detector high voltage electrode e17 in the r-z-plane (a)
at the FRM II for U17 = −30 kV and (b) at the ILL for U17 = −15 kV. To avoid
electrical breakdown initiated by field emission, in the new design of the high voltage
electrode e17, all edges (gray) were carefully rounded and electropolished (see also
Fn. 11). The comparison between (a) and (b) shows that the electric field strength
could be reduced by at least a factor of 3. Please note the different color ranges
in (a) and (b). We note that, compared to all other electric field calculations with
COMSOL 3.2b Multiphysics shown in this thesis, Figure (b) stems from a calculation
with the new version COMSOL 3.5 [187].
replaced by two half cylinders. The comparison between Figs. 3.13 and 3.14 shows that,
in the new design of the high voltage electrodes e16 and e17, the potential Penning traps
shown in Fig. 3.13 are no longer present.
The High Voltage Electrode
As discussed earlier on page 56 under “Penning Traps and Penning Discharge” and in the
previous section, also the detector high voltage electrode e17, shown in Fig. 3.12b, was
changed in shape compared to our first beam time at the FRM II. The proton detector has
to be shielded from radioactive background, mainly gamma’s from the neutron apertures
and from neighboring instruments. Therefore, the detector (described in Sec. 3.3.1) is
surrounded by a stainless steel tube which has a massive cup on its bottom (see also
Fig. 3.20b). To avoid electrical breakdowns, all edges of the tube were carefully rounded
and electropolished11, and the screws connecting the insulator and the preamplifier tube
were covered with two semicircular, rounded, and electropolished pieces (see Fig. 3.20b
for details). In combination with a reduced acceleration potential, U17, the electric field
strength could be reduced by at least a factor of 3, as can be seen from Fig. 3.15.
Details on the software used for axisymmetric magnetic and electric field calculations,
with zonal harmonic expansion, as well as for relativistic charged particle tracking, with an
8th order Runge-Kutta method, are found in [47, 188–190] (see also [191–195]). For non-
axially symmetric applications, as, e.g., the grid electrode e1, the quadrupole electrode e2,
11Please note that in the first half (November/December 2007) of our latest beam time parts of the
surface have been scratched. The scratches were removed only by mechanical polishing.
62 CHAPTER 3. THE ASPECT EXPERIMENT
the DV electrode gr, the lower and upper dipole electrodes e8 and e16, and the rounded
detector cup, COMSOL 3.2b Multiphysics Electromagnetics Module was used [196].
3.2.2 The Superconducting Coil System
The aSPECT spectrometer consists of a system of eleven12 superconducting coils, denoted
as c1 to c11 (see also Fn. 5), placed inside a cylinder with a length of three meters and a
diameter of seventy centimeters. The design of the (superconducting) coil system is shown
in Fig. 3.7. The coil system and its magnetic field are axially symmetric. The aSPECT
magnet generates a strong magnetic field which varies from 0.6 to 6T along the symmetry
axis, and down to 5Gauss in a radial distance of five meters from the DV (see App. A for
details), both at the design current of Imain = 100A.
The magnetic field of the aSPECT magnet came close to its design shape, but still
the superconducting correction coils c3 and c5 and the normal conducting correction coils
c12 and c13 were necessary to correct for small deviations both in the DV and the AP,
respectively (for details see [168]). All superconducting coils are connected in series and
operated in persistent mode13, except for the correction coils c3 and c5. In this way,
the ratios of the magnetic fields in the DV, the AP, and also at the height of the proton
detector remain constant in time, independent of the applied current Imain. We mention
that, in persistent mode, the values of the magnetic field B0 and BA in the DV and the
AP, respectively, as well as the magnetic field ratio rB = BA/B0 decrease very slowly
with time. The latter as the correction coils are not operated in persistent mode, i.e., the
magnetic field ratio rB changes with time by [53]:
1 ∂rB = −1.1× 10−4 year−1 . (3.40)
rB ∂t
The magnetic field along the z-axis of the aSPECT spectrometer is shown in the top
of Fig. 3.8. The coils c1 to c13 generate a very homogeneous magnetic field both in the
DV and the AP, ensuring a relative accuracy of the magnetic field ratio rB in the order
of 1 × 10−4 (for details see Sec. 4.1.4 and Ref. [53]). In the region of the lower dipole
electrode e8, the coil system produces also a rather homogeneous magnetic field. For this
reason, we can apply a potential difference14 of up to 3 kV between sides R and L without
any significant influence on the adiabatic proton transport between the DV and the AP
[47].
In the DV, the tiny correction coils c3 and c5 serve to adjust the shape of the magnetic
field; they can provide a magnetic field of up to 1% of the main field. We apply a slight
magnetic field gradient there, with the field decreasing towards the proton detector, as
shown in Fig. 3.16a. In this way, no protons can be trapped by the magnetic mirror effect
between the DV and the electrostatic mirror, ensuring 100% acceptance for decay protons.
In the AP, two pairs of (external) correction coils serve to adjust the shape of the
magnetic field. The pair c12 and c13 is operated in anti-Helmholtz configuration to correct
12Please note that the coils c2a to c2c as well as c4a to c4c are interpreted as one superconducting coil,
although they are manufactured in three parts each with a different cross-section of the windings.
13In persistent mode, the windings of a superconducting magnet are short-circuit with a piece of su-
perconductor, a so-called persistent switch, as soon as the magnet has been energized. Then its windings
become a closed superconducting loop and the power supply can be turned off. The great advantage of
this method is the high stability of the magnetic field, independent of possible fluctuations of the power
supply.
14Assuming that one side of the lower dipole electrode e8 is connected to ground.
3.2. THE RETARDATION SPECTROMETER ASPECT 63
(a) (b)
Figure 3.16: The values of the magnetic field along the z-axis of the aSPECT spectrometer in (a)
the decay volume (DV) and (b) the analyzing plane (AP). The slight magnetic field
gradient in the DV ensured that no protons can be trapped by the magnetic mirror
effect between the DV and the electrostatic mirror. Due to the local maximum in the
AP the adiabatic transmission condition Eq. (3.24) is automatically fulfilled in the
region close to the AP. Input data for the magnetic field calculation: Imain = 100A,
I3 = 50A, I5 = 21.4A, I12 = −I13 = 36.4A, I14 = I15 = 0.
for small deviations from the design shape; the pair c14 and c15 in Helmholtz configuration
to change BA by up to 1%. In comparison with our first beam time at the FRM II, the
coils c14 and c15 were added and subsequently the distance and the position of the coils c12
and c13 changed (compare with, e.g., Figs. 2.14 and 4.2). Changing BA by 1% allows us to
quickly perform a statistically significant test of the above calculated transmission function
Eq. (3.30), cf. Sec. 5.5.4. Therefore, the Helmholtz coils have to be placed symmetrically
around the maximum of the electrostatic potential in the AP. Consequently, the anti-
Helmholtz coils were mounted around the Helmholtz coils, as generally Helmholtz coils
have a shorter di√stance. But, due to limited√space, the anti-Helmholtz coils were installed
at a distance of 2R instead of the usual 3R, where R is the radius of the coils. The
typical current settings are
Imain = 70A, I3 = 35A, I5 = 15A, I12 = −I13 = 25.5A, and I14 = I15 = 0. (3.41)
Figure 3.16b shows that the magnetic field exhibits a local maximum, BA, in the AP,
with a relative uncertainty in the order of 1 × 10−4. In the region close to the AP, the
electrostatic potential is nearly uniform. For, e.g., UA = 400V, the potential in the region
close to the AP is only by 2mV smaller than in the center of the AP. If the magnetic field
would have a minimum instead, the minimum therefore would have to be extremely flat
to fulfill the adiabatic transmission condition Eq. (3.24). On the other hand, for our local
maximum, we get
UA − U0 > U(z)− U0 and (3.42)
B −B(z) sin20 (θ0) > B 20 −BA sin (θ0) (3.43)
in the region close to the AP, i.e.,
B0 −B(z) sin2(θ0) U(z)− U
> 1 0> . (3.44)
B0 −BA sin2(θ0) UA − U0
64 CHAPTER 3. THE ASPECT EXPERIMENT
Then the adiabatic transmission condition Eq. (3.24) is automatically fulfilled in the region
close to the AP:
B(z)
T ad|| (z) = Ttr(θ0)− e (U(z)− U )− T (θ
2
0 tr 0) sin (θ0)
B0
e (UA − U0) − − − B(z) e (U − U )= e (U(z) A 0U 2
− B 2 0
) sin (θ0)
1 A B BAB sin (θ0) 0 1− B sin
2(θ0)
0 0
− B0 −B(z) sin
2(θ
= ( ) 0
)
e UA U0 2 − e (U(z)− U0)B0 −BA sin (θ0)
(3.44)
> 0. (3.45)
In the detector region, the magnetic field is increased by about a factor of 2 compared
to the DV, as can be seen from Fig. 3.8. The smaller cross-section of the flux tube therefore
allows us to use a smaller proton detector. To ensure that no decay protons are reflected
by the magnetic mirror effect, the protons are accelerated by the high negative potential
of the electrodes e16 and e17. At the same time, the magnetic mirror effect significantly
reduces the electron background (see also Sec. 3.4.4).
For the latest beam time, the aSPECT spectrometer had to move to the Institut Laue-
Langevin (ILL) in Grenoble, France. In order not to disturb neighboring experiments, in
particular the spin-echo spectrometer IN11, by the fringe field of the strong aSPECT
magnet, a magnetic field return was built. The magnetic field return is shown in Fig. A.5
in App. A. It reduces the exterior magnetic field to less than 1Gauss in a radial distance
of five meters, as can be seen from Fig. A.6 in App. A. At the same time, the magnetic
field return does not affect the homogeneity of the internal magnetic field, cf. Fig. A.7 in
App. A. Details on the magnetic field return are found in App. A and in Ref. [1].
3.3 The Detection System
The aSPECT data acquisition (DAQ) consists of several parts: The proton detector, the
signal processing electronics, and the DAQ computers and slow control. Details are found
in Refs. [51, 52].
3.3.1 The Proton Detector
The main part of the aSPECT DAQ is the proton detector. In neutron beta decay, the
proton is emitted with a maximum kinetic energy of only 751 eV. To obtain a signal well
separated from the noise decay protons are usually post-accelerated onto the detector by
at least −25 kV.
Our first measurements at the FRM II were performed with a silicon PIN diode with
25 strips, with a size of 25 × 0.8mm2 each, set at a potential of about −30 kV [33]. In
such PIN diodes, the PN-structure of a normal diode is separated by an intrinsic layer
to increase the active volume of the detector. However, the capacitive noise of such a
detector is proportional to its active area. To implement a large detector surface, the
detector therefore has to be divided into many single active areas with a separate readout
for each area. But even with our high acceleration potential the proton signal was hardly
separated from the noise of the system [49, 197] (see also Fig. 2.15a).
3.3. THE DETECTION SYSTEM 65
(a) (b)
Figure 3.17: Working principle of a silicon drift detector (SDD). (a) Schematic cross-section, taken
from Refs. [51, 199]. (b) Calculation of the potential distribution inside a SDD
detector with typical values. For better visibility of the valley, the axis of the potential
is plotted inverted. Calculation taken from M. Simson [52]. Ionizing radiation enters
the detector through the back contact. Free electrons created within the potential
valley drift towards the central annular anode. The anode is connected to the gate
(G) of the integrated FET which acts as a first amplification stage.
Repeated electrical breakdown during beam times triggered to go for a different so-
lution. In our latest beam time at the ILL, we used a more sophisticated semiconductor
detector. In our so-called silicon drift detector (SDD), supplied by PNSensor [198], the
thermal noise is decoupled from the active area.
For more details on semiconductors and semiconductor detectors, the reader is referred
to standard textbooks [199, 200].
Working Principle of a Silicon Drift Detector
The principle of a SDD was first proposed by Gatti and Rehak in 1984 [201] and is based
on the sidewards depletion principle. It allows the full depletion of a large detector volume
with a very small readout node. This greatly decreases the thermal noise compared to a
conventional PIN diode.
Figure 3.17a shows a schematic cross-section of a SDD. The bulk material of the
detector is n− doped silicon15, with a smooth p+ layer on one side and concentric p+ rings
on the other side. The back contact is connected to a positive voltage, whereas the rings
on the front side have different potentials. The n− doped regions between these rings
have a rather high resistivity and therefore act as a voltage divider. The corresponding
potential valley is shown in Fig. 3.17b. Ionizing radiation enters the detector through the
back contact. Free electrons created inside the potential valley drift towards the center of
the front side where they are collected by a small n+ doped anode.
The Detector Chip
In our experiment, three SDDs are used, implemented in a row on one silicon chip. Fig-
ure 3.18a shows the entrance side of the detector which is covered with a protective layer
15The superscript − refers to a faint degree of doping, whereas a + indicates a strong doping.
66 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.18: Detector chip used at the ILL. (a) Photograph of the entrance side. Each of the
three SDD pads has an active area of 100mm2. The chip is supplied on a special,
ultra-high vacuum (UHV) suitable ceramics board. To establish an electrical contact
wires are bonded between the chip and the board. The ceramics board is inserted
in two commercially available 20 socket zero-insertion-force connectors which are not
suitable for UHV. For use inside the spectrometer a special holder was designed [52].
(b) Sketch of the electromagnetic set-up. For better visibility, only non-axisymmetric
electrodes are shown. Neutrons are coming from the left and pass through the decay
volume (red), where only a few decay. The decay products are guided by the strong
magnetic field towards the detector. The three detector pads, denoted by 5, 6, and 7,
are aligned parallel to the neutron beam. Protons emitted in the negative z-direction
are reflected by the electrostatic quadrupole (black). Trapped protons are removed
from the flux tube with the lower dipole electrode (turquoise). The upper dipole
electrode (purple) serves to align the neutron beam on the detector. Note that the
electrodes are scaled to their projection by the magnetic field onto the detector.
of 30 nm of aluminum. Each of the three detector pads has an active area of 100mm2 in
the form of a square with a side length of 10.3mm and rounded corners with 2mm radius.
The total size of the chip is 34× 14× 0.45mm3.
One specific feature of our SDD is that the first amplifying field-effect transistor (FET)
is integrated directly onto the detector. This minimizes the cable length from the detector
to the first amplification stage to nearly zero and hence decreases both the capacitive noise
and the pick-up of external noise compared to conventional detectors. Another specific
feature is that temperature sensors are implemented on the detector chip.
With the new proton detector, the acceleration potential could be reduced by a factor
of 2 down to −15 kV [51, 52]. This solved the problem of frequent electrical breakdowns
and, in combination with the redesign of several electrodes (see Sec. 3.2.1 for details),
significantly improved the background conditions (discussed in Sec. 5.3). As can be seen
from Fig. 3.19, the proton signal is sufficiently separated from the noise.
Mechanical Set-up
For our measurements, the detector has to be placed in the high magnetic field that guides
and focuses the decay protons onto the detector. For reasons of maintenance, however, the
detector should be rather easily accessible. Hence, we use a retractable system, in which
the detector can be moved into and out of the magnet. The cold bore tube of the aSPECT
3.3. THE DETECTION SYSTEM 67
(a) (b)
Figure 3.19: Pulse height spectra measured at (a) the FRM II with the silicon PIN diode set at
a potential of −30 kV and (b) the ILL with the SDD detector set at −15 kV. With
increasing barrier voltage the count rate in the proton peak (right peak) decreases
whereas the electronic noise (left peak) is not influenced. The direct comparison
shows that the noise could be sufficiently separated from the proton signals. The lower
acceleration potential significantly improved the background conditions (proton-like
peak).
magnet is separated from the detector vacuum by means of an ultra-high vacuum (UHV)
gate valve. Thus the detector can be maintained without venting the cold bore tube.
Figure 3.20a shows the mechanical set-up of the detector. The mechanics can be
divided into two parts: The lower part is on high voltage, whereas the upper part stays on
ground potential. These two parts are electrically separated by a non-magnetic ceramics
tube. The high voltage part, shown in Fig. 3.20b, consists of two nested tubes: The
inner tube houses the preamplifier at atmospheric pressure. The outer tube surrounds the
detector which is mounted outside the closed lower end of the inner tube. To shield the
detector from radioactive background, mostly gamma’s from the neutron apertures and
from neighboring instruments, the tube has a massive stainless steel cup on the bottom.
The digital electronics is mounted on top of the spectrometer, inside an aluminum box
for electrical shielding. The aluminum box, in turn, is situated inside a bigger perspex
box for protection against high voltage. The cables which connect the analog and digital
electronics are contained in an acrylic glass tube. To shield the cables against electrostatic
noise they are surrounded by a stainless steel tube that is connected to the high voltage.
Two pressurized air tubes are used to cool the preamplifier board and the digital
electronics inside the aluminum box, respectively. Since the air from the first tube escapes
through the central cable tube no water can condense on the inside of the ceramics tube.
The ceramics tube, in turn, is cooled passively by the surrounding cold bore walls. We
note that water on the inside of the ceramics tube could cause electrical breakdowns.
68 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
Figure 3.20: (a) A detailed sketch of the mechanics with the detector inside the magnet. A
membrane bellows is connected to the ground potential side of the insulator. It
separates the inner tube at atmospheric pressure from the UHV inside the magnet and
enables the movement of the detector. The digital electronics is mounted on top of the
set-up inside an aluminum box. The upper E×B electrode is also inserted from the
top. (b) A detailed sectional drawing of the high voltage part consisting of insulator,
preamplifier tube, detector, and shielding cup, taken from [52]. The detector is
directly connected to the preamplifier by a 50 pin UHV SUB-D feedthrough. The
amplified signals are transferred by coaxial cables in the central tube to the aluminum
box.
3.3.2 Signal Processing Electronics
Amplification Boards
The preamplifier board is directly connected to the detector feedthrough, as can be seen in
Fig. 3.20b. In this way the shortest possible distance between detector and amplification is
achieved, resulting in low electronic noise. This board also distributes the several voltages
needed to operate the detector. In addition, a readout circuit for the temperature sensors
on the detector chip is implemented on the preamplifier board.
The amplified signals are transferred to an adapter board inside the aluminum box on
top of the spectrometer. This board shapes the signals and transfers them to the analog
to digital converter (ADC). The raw signal shape consists of a steep rise followed by a long
exponential decay. The shaper is mostly sensitive to the rise and shortens the pulse so that
it can be further processed by the digital electronics. The adapter board also distributes
3.3. THE DETECTION SYSTEM 69
Figure 3.21: Illustration of the trigger algorithm. A typical proton event is shown in blue. The
algorithm compares the level of the baseline to the signal height. Window w1 (red)
is used to determine the baseline. If the mean value of window w2 (green) differs by
more than a given threshold from the baseline, the trigger condition is fulfilled. Both
windows are separated by the window distance. The exact stored region around the
event is given by the length of the event window and the trigger buffer.
the voltages from the voltage divider board to the different parts of the electronics.
Digital Electronics and the Trigger Algorithm
The last part of the signal processing electronics is the sampling analog to digital con-
verter (sADC) board, also mounted inside the aluminum box. The shaped signals are
continuously digitized by a 12 bit ADC. Its sampling frequency is 20MHz resulting in
time bins with a width of 50 ns. This frequency offers a reasonable compromise between
time resolution and data volume as the events typically have a length of about 5µs.
Since there is no external trigger in our experiment the digitized signals are contin-
uously sampled by field programmable gate arrays (FPGAs) and analyzed with respect
to a trigger algorithm: The output signal of the ADC is continuously shifted through a
register and discarded at the end if no pulse was detected. As shown in Fig. 3.21, the
general algorithm is based on the comparison of two windows, denoted as w1 and w2,
within the shift register. Window w1 is used to determine the baseline of the signal. The
two windows are separated from each other by the window distance. If the mean values of
the two windows differ by more than a given threshold, the trigger condition is fulfilled.
To suppress triggers on noise an additional parameter, termed as delay, is introduced,
so that the threshold condition has to be fulfilled several times in a row before a trigger
decision is made. To record the entire pulse a certain amount of ADC values before the
event, denoted as buffer, is stored. The total amount of stored ADC values is given by
the length of the event window. Furthermore, a time stamp is added to the event.
The lengths of the event window, the two windows w1 and w2, and the trigger buffer
as well as the delay can be set from the control program, whereas the window distance
is fixed to 24 time bins (0.8µs). During most of our measurements at the ILL, the event
window was set to a length of 100 time bins (5µs), both windows w1 and w2 to 24 time
70 CHAPTER 3. THE ASPECT EXPERIMENT
bins (0.8µs), the trigger buffer to 15 time bins (0.75µs), and the delay was set to 3.
If a trigger decision is made, the FPGA attaches a header16 with additional information
(heartbeat, slow-control, overflow, and ADC channel and size of the event) about the pulse.
Then the event is sent to the DAQ computer, where it is stored for offline data analysis.
The DAQ computer is set at ground potential. To operate the sADC board at the detector
high voltage, the data is therefore transfered via a HOTLink interface17 with optical fiber
cable.
Readout Software and Instrument Control Set-up
When a measurement file is completed, the raw data is converted, i.e., decoded to a
parseable data format by a C/C++ program. Since our data analysis software is based on
ROOT18 the decoding program creates a ROOT tree with all important information about
every event from the raw data file19. In particular, for each event the program determines
a baseline by the mean value of the first 15ADC values and a preliminary pulse height by
the maximum difference between the signal and its baseline. In addition, for each file the
decoding program creates a pulse height histogram.
For our measurements, several voltages and other values have to be set and monitored.
Thus we have used five personal computers (PCs) for data taking and instrument control:
the DAQ PC (controls and reads the data from the sADC), the decode PC (decodes the
data from the detector), the data server (used to store the data from the detector), the
magnet control PC (reads the temperature sensors of the magnet and serves to set and
monitor the high voltages on the detector and on one side of the upper E×B electrode),
and the control PC (sets and reads the AP voltage, reads the neutron counter, controls
the movement of the neutron shutter, writes several log files, and starts and stops the
different operations performed by the other PCs). For control and readout of the different
devices several LabVIEW20 programs on the different PCs are used. The different PCs
communicate with each other via DataSocket21.
Further details on the detector, the signal processing electronics, simulations and in-
vestigations of detector properties, and the used programs can be found in Ref. [52].
3.4 Systematic Effects
The ultimate goal of the aSPECT experiment is to improve the uncertainty in the neutrino-
electron correlation coefficient a to 0.3% [47]. In order to meet this target figure, all
systematic effects must be kept below ∆a/a = 0.1%. The dominant systematic effects are
thoroughly discussed in Refs. [46, 47]. In this section, we will only summarize the main
points.
16For details on the data structure see Refs. [52, 197].
17Interface to transfer data over long distances with serial links.
18Object oriented framework for large scale data analysis developed at CERN [202].
19For details on the structure of the ROOT tree see Ref. [52].
20Graphical programming environment developed by National Instruments (NI).
21Standard to transfer data via the network developed by NI.
3.4. SYSTEMATIC EFFECTS 71
Figure 3.22: Relative change of the angular correlation coefficient a for different magnetic field
gradients, ∂B0/∂z, in the decay volume (DV), assuming a uniform neutron beam
density. For better visibility, the y-axis is broken at ∆a/a = 1.6%. An inhomogeneity
of (Bmax −Bmin) /B0 = 8× 10−4 over the full height, hn-beam, of the neutron beam
corresponds to a shift in a of ∆a/a = −0.001(50)%. Input data for the simulation
(in INM approximation): BA/B0 = 0.203, UA = 50, 250, 400, 500, 600V, and the
recommended value for a = −0.103 [10]. The error bars, in the order of 0.05%,
represent the error by fitting Eq. (3.32) to the simulated proton count rates only.
3.4.1 The Adiabatic Transmission Function
As discussed earlier in Sec. 3.1.2, the transmission function Eq. (3.30) depends only on
the ratio rB of the magnetic fields in the AP and the DV and the barrier voltage UA. As
discussed further in Sec. 3.1.3 (see also Fig. 3.5), we have to know the magnetic field ratio
rB with a relative accuracy of 1× 10−4 and the potential barrier UA with an accuracy of
better than 10mV, in order to keep systematic uncertainties in a below ∆a/a = 0.3%.
In order to know the magnetic field ratio rB = BA/B0 with a relative accuracy of
∆r /r = 1 × 10−4B B , we have to know both the magnetic fields in the AP and the DV
with a relative accuracy of ∆BA/BA = 1 × 10−4 and ∆B0/B0 = 1 × 10−4, respectively.
Hence, a high temporal stability is required, as it was not yet possible to measure the
magnetic field during data taking. First tests with a nuclear magnetic resonance (NMR)
system with polarized 3He (developed by F. Ayala Guardia [53]) have shown that the
magnetic field ratio rB is stable within ∆rB/rB = 0.1× 10−4 for 1month (cf. Eq. (3.40)),
well within our tolerance of 1 × 10−4. We mention that in the upcoming measurements
with aSPECT (see also Sec. 7.3) an improved NMR system with 1:1 acetone and ethanol
mixture (developed by F. Ayala Guardia [53]) will be installed close to the DV and the
AP.
However, the magnetic field in the DV exhibits a slight magnetic field gradient of
(∂B /∂z) /B = −1 × 10−4 cm−1,220 0 as can be seen from Fig. 3.16a. This corresponds
to an inhomogeneity of the magnetic field in the DV of (Bmax −Bmin) /B0 = 8 × 10−4
over the full height of the neutron beam (see Sec. 4.1.3 for details), out of our tolerance
22Please note that the figure slightly deviates from that given in Ref. [53], as the final analysis of the
magnetic field measurements had not been completed at the time of this analysis.
72 CHAPTER 3. THE ASPECT EXPERIMENT
of 1 × 10−4. Consequently, we have to weight the transmission function Eq. (3.30) with
the neutron beam density n(P0). Figure 3.22 shows the relative change in a for different
magnetic field gradients, assuming a uniform neutron beam density. The current magnetic
field gradient of (∂B0/∂z) /B0 = −1× 10−4 cm−1 corresponds to a shift in a of
∆a/a = −0.001(50)%. (3.46)
Hence, the influence of the present magnetic field gradient on the neutrino-electron cor-
relation coefficient a is negligible at the level of 0.1%. Further investigations for different
shapes of the neutron beam profile are found in Ref. [53].
According to our electric field calculations presented in Sec. 3.2.1, the electrostatic
potential within the DV is smaller than 1mV and its variations at the AP are smaller
than 2mV, both well within our tolerance of less than 10mV. However, the following
effects may influence the potential barrier UA:
Negative space charges: The accumulation of negative space charges around the AP
could change the electrostatic potential UA. Experiences from similar experiments
[41, 174, 203, 204] suggest that the accumulation rate is sufficiently small.
Work function differences: Work function differences of the different materials should
cancel, as all electrodes are gold-plated and kept at the same temperature of about
70K (of the cold bore tube of the aSPECT magnet), except for the high voltage elec-
trodes e16 and e17. However, a variation of the work function of up to 250meV was
found in a cylindrical sample electrode at atmospheric pressure and room tempera-
ture, as can be seen from Fig. 6.1. The influence of possible variations of the work
function within an electrode or between different electrodes is thoroughly discussed
in Chap. 6.
Surface charges: Surface charges on a possible insulating surface layer of the electrodes
could influence the electrostatic potential UA. In Ref. [205] measurements of this
effect due to external radiation are presented for different surfaces. Extrapolation
of these results suggests that this effect is negligible.
In order to measure in situ the potential difference UA − U0 between DV and AP, a
monochromatic ion source is under development. Key requirements on the calibration
source are:
• operation in a strong magnetic field and
• provision of charged particles with known energy and energy spread of about 10meV.
A photo-electron source turned out not to fulfill these requirements [3]. First tests with a
calibration source in which monoenergetic 4He ions are produced are found in Ref. [34].
3.4.2 Non-Adiabatic Proton Motion
The aSPECT experiment was designed to eliminate the shortcomings of its predecessor [41]
(see also page 37 under “The Measurement of Byrne et al. at the Institut Laue-Langevin”),
in particular such that effects of to non-adiabatic proton motion are negligible [47]. As
stated earlier in Sec. 3.1.2, in the adiabatic approximation, the orbital magnetic moment
µ = Tp,⊥/B is constant and the transmission function can be calculated analytically.
3.4. SYSTEMATIC EFFECTS 73
Table 3.2: Relative change of the angular correlation coefficient a for different magnetic field values
B0. With decreasing magnetic field the non-adiabatic change of the orbital magnetic
moment increases exponentially. For magnetic field values above B0 = 1.5T the influ-
ence of non-adiabatic proton motion on a is negligible, while substantial for magnetic
field values below B0 = 1T. Input data for the simulation [47]: magnetic field ratio
B0/BA = 5 and acceleration potential |U8R − U8L| = 3 kV fixed (see also Fn. 14).
B0 [T] ∆a/a [%]
3 0.001
1.5 0.04
1.2 0.5
0.9 4
0.6 20
Our proton trajectory calculations showed that the value of µ in the AP is indeed
close to its value at the decay point P0. In contrast, in regions with high electric and/or
magnetic field gradient, e.g., at the lower dipole electrode e8, the value of µ oscillates
with a period equal to the local gyration period; where the oscillation amplitude increases
with the field gradients. Hence, the decay protons seem to remember their initial orbital
magnetic moment when passing from one homogeneous field region to another [176] (here:
from the DV to the AP). However, our trajectory calculations showed that breakdown
of the adiabatic approximation sets an upper limit of 3 kV for the potential difference
between sides R and L of the lower dipole electrode e8 (see also Fn. 14). With increas-
ing acceleration potential the proton’s gyration length and hence the deviation from the
adiabatic approximation increase.
The effect of non-adiabatic proton motion is more severe for those decay protons that
cannot pass the potential barrier and are subsequently trapped between the electrostatic
mirror and the AP. These protons perform several axial oscillations between the electro-
static mirror and the AP, i.e., several passages through the lower dipole electrode e8,
before they are removed from the flux tube with the dipole electrode. On the one hand,
the non-adiabatic change of the orbital magnetic moment increases with the number of
these oscillations [41]; on the other hand, the number of these oscillations decreases with
increasing potential difference. For our upper limit of 3 kV for the potential difference (see
also Fn. 14), the trapped protons perform less than five to six axial oscillations [47].
According to Ref. [177], the non-adiabatic change of the orbital magnetic moment
increases exponentially with decreasing magnetic field. In Table 3.2 we list the relative
change in the neutrino-electron correlation coefficient a for different values of the magnetic
field B0, for our upper limit of 3 kV for the potential difference (see also Fn. 14). For
magnetic field values above B0 = 1.5T the influence of non-adiabatic proton motion on
a is negligible. To verify our calculations, measurements with different heights of the
magnetic field were investigated (discussed in Sec. 5.5.5).
3.4.3 Residual Gas
A decay proton should reach the proton detector if and only if the adiabatic transmission
condition Eq. 3.24 is fulfilled at the AP. On their way from the DV to the proton detector,
74 CHAPTER 3. THE ASPECT EXPERIMENT
Table 3.3: Critical pressure values of elastic p-H2 scattering for different acceleration potentials
|U8R − U8L| [47] (see also Fn. 14).
|U8R − U8L| [kV] pcr [mbar]
3 5× 10−8
0.3 1× 10−8
0.03 1.4× 10−9
these protons may interact with residual gas molecules and hence change their energy and
direction. Three cases must be considered:
• change of energy and direction due to elastic scattering,
• change of energy and direction due to inelastic scattering, and
• neutralization due to charge exchange processes.
The systematic effect due to these collisions is proportional to the residual gas density, i.e.,
for a fixed temperature (here: of about 70K of the cold bore tube), to the gas pressure.
Thus, we define pcr as the critical value of the residual gas pressure23 at which interactions
introduce a systematic effect on a at the level of ∆a/a = 0.1%. In the following, we report
the results of Monte Carlo (MC) simulations for the above mentioned three processes [47]:
Elastic scattering: Elastic scattering processes may change kinetic energy and pitch
angle24 of the decay protons. Two cases occur:
• trapped protons with Tmintr < T0 < Ttr(θ0) can be transmitted through the AP,
• protons with T0 > Ttr(θ0) might be hindered to pass the potential barrier.
The scattering probability is proportional to the particle’s path length. Thus, the
first effect on the trapped protons is more important, as they perform several axial
oscillations between the electrostatic mirror and the AP before they are removed
from the flux tube with the lower dipole electrode e8. In Table 3.3 we list the
critical pressure values of elastic p-H2 scattering for different settings of the dipole
electrode. With increasing potential difference the trapping time decreases and hence
the critical pressure increases.
To verify our MC simulations, measurements with different settings of the lower
dipole electrode e8 were investigated (discussed in Sec. 5.3.2).
Inelastic energy loss: Inelastic scattering processes may also change kinetic energy and
pitch angle of the decay protons. Compared with elastic scattering processes, the
change of the pitch angle is negligible [47] (see also [206–210]). The energy loss, in
turn, is dominated by rotational and vibrational excitations. For inelastic scattering
of protons on hydrogen, the energy loss due to vibrational excitations sets an upper
limit of
p −8cr = 4× 10 mbar. (3.47)
23Please note that all figures in this section are based on a temperature of 60K (see also [47]).
24Compared with the (initial) polar angle, the pitch angle denotes the angle between the proton’s
momentum and the magnetic field.
3.4. SYSTEMATIC EFFECTS 75
Table 3.4: Critical pressure values of charge exchange processes for different residual gases [47].
Gas pcr [mbar]
H2 2× 10−8
Ar 1× 10−8
N 2× 10−82
O 4× 10−82
He 1× 10−6
For inelastic scattering of protons on other atoms and molecules, this has to be part
of further investigations (by means of MC simulations).
Charge exchange: A decay proton may capture an electron from a residual gas molecule:
p + M −→ H + M+. (3.48)
The resulting positive ion will have very low energy. Thus, if the process takes place
between the DV and the AP, no event will be detected. In Table 3.4 we list the
critical pressure values for different residual gases.
Altogether, for a residual gas pressure of 10−8 mbar or better, the influence of inter-
actions with residual gas molecules on the neutrino-electron correlation coefficient a is
negligible at the level of 0.1%. In comparison with our first beam time at the FRM II,
four getter pumps25 were installed directly on the electrode system to improve the vacuum
conditions. Two of the four getter pumps are shown in Fig. 3.9a. In this way, the residual
gas pressure was improved to at least26 8 × 10−9 mbar, i.e., by at least a factor of 2.5
compared to our first beam time at the FRM II [33].
3.4.4 Background
With regard to the background, we distinguish between:
Correlated background events: In neutron beta decay, the decay products, proton,
electron, and anti-neutrino, are emitted simultaneously. In the aSPECT experi-
ment, only protons and electrons can be detected. Decay electrons are detected
only if they were emitted towards the proton detector and if they can overcome the
electromagnetic mirror right in front of the detector. Due to their higher energy but
smaller mass, the time-of-flight (TOF) of the electrons is about one thousands of the
TOF of the decay protons. Thus, the detection time difference between correlated
coincidence events may be estimated by the TOF of the proton. The minimum
TOF of decay protons is about 5.2µs27, where the proton was emitted towards the
25To be specific, four SAES CapaciTorr D 400-2DSK getter cartridges were installed. In order not to
disturb the homogeneous magnetic field, the getter cartridges were chosen for their non-magnetic (relative
permeability µr ≈ 1.011), high performing St 172 (Zr-V-Fe) porous sintered getter material.
26In order not to disturb the homogeneous magnetic field, the cold cathode gauge was mounted on the
bottom flange of the aSPECT spectrometer, at the far end of a 2meters long, warm CF40 tube.
27Please note that in Refs. [34, 52, 211] the minimum TOF is misstated. The minimum TOF at UA = 0V
was already corrected in Ref. [33] to 5.2µs [47], for slightly different settings of the spectrometer.
76 CHAPTER 3. THE ASPECT EXPERIMENT
proton detector, opposite to its correlated electron. Hence, for a dead time of the
electronics smaller than the minimum TOF of the protons, i.e., τdead ≤ 5.2µs, all
correlated proton events should be detected (for details see Sec. 5.2.1). Indeed, in
our latest beam time at the ILL, a small fraction of the decay protons was lost due
to a problem in the detector electronics (discussed in Sec. 5.4.1).
Uncorrelated background events: Positive ions coming from the residual gas or the
electrodes, gamma radiation and high-energy electrons created by neutron capture
and by cosmic rays cause uncorrelated background events. Obviously, the uncor-
related background can be subdivided into beam-related and environmental back-
ground. Although beam-related background is often a problem in neutron decay ex-
periments, it is not treated differently in our data analysis (for details see Sec. 5.2.2),
provided that the background does not depend on the barrier voltage. As mentioned
earlier in Sec. 3.1, in our experiment, the background is measured at UA = 780V.
Further experimental studies are necessary to make sure that this method does not
change the background to be measured, by, e.g., applying the barrier potential to
different electrodes. Indeed, in our first beam time at the FRM II, the background
count rate showed a dependence on the barrier voltage (see also page 56 under
“Penning Traps and Penning Discharge” and Sec. 5.3.1).
3.4.5 Doppler Effect due to Neutron Motion
The motion of the decaying neutron also changes the observed energies and angles of the
outgoing particles relative to the energies and angles in the center-of-mass system (CMS)
of the neutron. The kinetic energy, TCMS, and polar angle, θCMS, of the outgoing particles
in the CMS system of the neutron c√hange√as follows:mp m
TLAB = TCMS + Tn + 2
p
TCMSTn cos θCMS and (3.49)
mn mn
sin θ
= arctan √CMSθLAB √ , (3.50)
cos + mθ p TnCMS mn TCMS
where TLAB and θLAB are the kinetic energy and polar angle of the outgoing particles in
the laboratory system, respectively, and Tn is the kinetic energy of the decaying neutron.
This momentum change, in turn, may affect the transmission function Eq. (3.30).
Cold neutrons have an average kinetic energy of about Tn = 4meV. To get a feeling
for the Doppler shift, we consider two examples:
(TCMS; θCMS) = (400 eV; 0°) −7 → (TLAB; θLAB) = (402.54 eV; 0°) , (3.51)
(TCMS; θCMS) = (400 eV; 30°) 7−→ (TLAB; θLAB) = (402.20 eV; 29.91°) . (3.52)
These are both enormous energy and angle changes, especially in comparison with the
required accuracy of 10meV.
In aSPECT, this effect is strongly suppressed due to the transverse proton detection
with respect to the neutron beam and its 4π acceptance (see also Sec. C.3). According
to Eq. (3.49), protons emitted opposite to the neutron motion have a laboratory energy
smaller than their CMS energy. We have therefore a large cancellation of Doppler effects.
The impact of the Doppler effect on the neutrino-electron correlation coefficient a was
calculated for a typical neutron velocity distribution and different settings of the barrier
potential [47]. For UA < 500V, the effect is smaller than 10−4. Hence, we do not expect
any essential systematic uncertainty from the Doppler effect.
3.4. SYSTEMATIC EFFECTS 77
3.4.6 Edge Effect
Depending on the size of the proton detector, the shape of the neutron beam profile (in
the DV) may induce another systematic effect. The neutron beam profile is projected
by the magnetic field onto the detector. The decay protons, in turn, are guided by the
magnetic field towards the detector, gyrating around a magnetic field line. As shown in
Fig. 3.23, at the edges of the detector, two cases occur:
• protons emitted outside the projected area can hit the detector and
• protons emitted within the projected area might miss the detector.
Depending on the shape of the neutron beam profile, two cases must be considered:
Uniform neutron beam profile: In the ideal case, the two above mentioned cases can-
cel, provided that the detection efficiency of the proton detector is uniform. We
mention that the cancellation is independent of the proton’s radius of gyration, as
can be seen from Figs. 3.23a and 3.23b.
Non-uniform neutron beam profile: In reality, the neutron beam profile is more in-
homogeneous. In this case, the probabilities for the two cases become different
and dependent on the proton’s radius of gyration, as can be seen from Figs. 3.23c
and 3.23d. This leads to an energy dependent systematic effect, as the proton’s
radius of gyration depends on both its emission angle and energy (cf. Eq. (3.3)).
We note that for a given definite radius of gyration, any proton emitted in-between
the two transition regions (see Figs. 3.23c and 3.23d) will hit the detector.
There are now two possibilities to eliminate or at least to minimize this effect:
• narrow the width of the neutron beam to the minimal width of the area in-between
the two transition regions, i.e., to the width of the proton detector minus four times
the maximum radius of gyration Eq. (3.5). For our typical current settings Eq. (3.41),
this would correspond to a width of 7.4mm, what in respect of statistics is not
acceptable.
• optimize the neutron apertures to obtain a neutron beam profile as uniform as
possible. We note that, if the neutron beam profile is not sufficiently homogeneous,
its shape has to be known precisely.
MC simulations have to be used to correct this effect, as the projection of the neutron
beam profile onto the detector is altered by both, the lower and upper, dipole electrodes.
To verify our calculations, measurements with different shapes of the neutron beam profile
were investigated (discussed in Sec. 5.5.6). The shape of each neutron beam profile was
measured by copper foil activations in front and behind the aSPECT spectrometer (for
details see Sec. 4.1.3).
3.4.7 Detection Efficiency
Even if a decay proton hits the detector it may stay undetected, due to the dead layer and
the response function of the proton detector. The detection efficiency, i.e., the probability
that a proton will be detected, depends on both its impact angle and energy. We note
78 CHAPTER 3. THE ASPECT EXPERIMENT
(a) (b)
(c) (d)
Figure 3.23: Illustration of the edge effect, for two different shapes of the neutron beam profile:
(top) uniform profile and (bottom) non-uniform profile. At the edges of the proton
detector, two cases occur: Protons emitted outside the projected area can hit the
detector, whereas protons emitted within the projected area might miss the detector.
The comparison between (a) and (b) shows that this two cases cancel, independent
of the proton’s radius of gyration; the comparison between (c) and (d) shows that
the probabilities for the two cases become different and dependent on the proton’s
radius of gyration.
that the average detection efficiency is irrelevant to our data analysis, as it just reduces
the count rates by a common factor.
The proton’s impact energy is given by its emission energy T0 ∈ [0, Tp,max] plus the
acceleration potential U17 ∈ [−15 kV,−10 kV], i.e., it will be close to the acceleration
voltage. Hence, the energy-dependent detection efficiency can be written as
fimpact energy ∝ 1 + c1 · T0, (3.53)
with a small constant c1. For |c1| < 200 ppmkeV−1, the influence of the energy-dependent
detection efficiency on the neutrino-electron correlation coefficient a is negligible at the
level of 0.1% [47]. We note that reasons for an energy-dependent detection efficiency are
• backscattering in the dead layer of the proton detector (discussed below) and
3.4. SYSTEMATIC EFFECTS 79
• the threshold to distinguish between proton signals and the electronic noise (see also
Sec. 5.4.1).
In the first case, a decay proton enters the detector, scatters several times inside the
detector, before it leaves the detector again. After a rather short time of about 0.5µs,
the proton will hit the detector again, because of the electromagnetic set-up of aSPECT.
This guarantees that at most one proton event is counted.
The angular-dependent detection efficiency, in turn, is dominated by backscattering in
the dead layer of the proton detector and can therefore be written as
∝ − cfimpact angle 1
2
, (3.54)
cos θp,det
where c2 is a small constant and √
B T sin2 θ
θp,det = θp,det(T0, θ0) = arcsin
det · 0 0 (3.55)
B0 T0 − e (U17 − U0)
is the angle between the proton’s momentum and the detector axis (cf. Eq. (3.18)). The
maximum impact angle, θp,det(Tp,max, 90°), of decay protons is 18° at U17 = −15 kV up to
22° at U17 = −10 kV. For c2 < 7× 10−3, the influence of the angular-dependent detection
efficiency on the angular correlation coefficient a is negligible at the level of 0.1% [47].
Investigations of the backscattering of decay protons are found in Ref. [52]. Figure 3.24
shows the probability of backscattering for different proton impact angles and energies
[52]. The fraction of backscattered protons is about 1.0% at U17 = −15 kV up to 1.8% at
U17 = −10 kV [52]. Here, we only present the possible influence of backscattering of decay
protons on the neutrino-electron correlation coefficient a, derived from this data28:
∆a/a < +0.22(16)% (3.56)
In fact, a backscattered proton might loose enough energy in the active volume of the
proton detector to be detected. Thus, Eq. (3.56) only sets an upper limit on the correction
of the backscattering of decay protons.
For details on simulations29 and investigations of detector properties the reader is
referred to the thesis of M. Simson [52].
28Further input data for the MC simulation (in INM approximation, with Coulomb correction
Eq. (2.56)): B0 = 2.177T, BA/B0 = 0.203, Bdet = 4.340T, UA = 50, 250, 400, 500, 600V, U17 = −15 kV,
number of generated events = 109, and a = −0.105 (derived from λ = −1.2701(25) [10]).
29The detector properties were simulated with the software package SRIM [212].
80 CHAPTER 3. THE ASPECT EXPERIMENT
Figure 3.24: The probability of backscattering of decay protons for three different impact angles
and energies. The fraction of backscattered protons is about 1.0% at U17 = −15 kV
up to 1.8% at U17 = −10 kV [52]. 6 × 105 protons were simulated for each point.
Data from Ref. [52].
Chapter 4
Measurements at the ILL
The aSPECT experiment was set up from November 2007 to June 2008 at the cold neu-
tron beam facility PF1B of the Institut Laue-Langevin (ILL) high-flux reactor in Grenoble,
France. The primary emphasis of this beam time has been put on the identification and
investigation of possible systematic effects with sufficient statistical accuracy. This is be-
cause background instabilities due to particle trapping and the electronic noise level of the
proton detector had prevented us from presenting a new value for the neutrino-electron cor-
relation coefficient a from our first beam time at the Forschungs-Neutronenquelle Heinz
Maier-Leibnitz (FRM II) in Munich, Germany [33, 49]. To avoid these problems, the
proton detector was replaced by a silicon drift detector (SDD) at significantly reduced ac-
celeration potential (discussed in Sec. 3.3.1), parts of the electrode system were redesigned
(see Sec. 3.2.1 for details), and the ultra high vacuum (UHV) conditions were improved
(for details see Sec. 3.4.3). As shown earlier in Fig. 3.19b (see Sec. 3.3.1), the electronic
noise could be sufficiently separated from the proton signal. The secondary focus of this
beam time was on the determination of a new value for the correlation coefficient a with
a total relative error well below the present literature value of 4% [10] (see also [40, 41]
and Sec. 2.4).
In this chapter, we give a brief overview of our measurements during the beam time at
the ILL. A detailed analysis of the measured data follows in Chap. 5. Details on detector
and electronics tests, measurements of the neutron beam profile, and measurements of the
magnetic field can be found in the theses of M. Simson [52], M. Borg [34], and F. Ayala
Guardia [53], respectively.
4.1 Experimental Set-up at the ILL
Our experiment makes use of cold neutrons, i.e., neutrons with kinetic energies from
0.5µeV to 0.025 eV (see also Fn. 38 in Chap. 2). Normally these neutrons are produced
in a research reactor or by a spallation source. The Institut Laue-Langevin (ILL) is an
international research center that operates the most intense neutron source in the world.
The ILL’s high-flux reactor is a fission reactor1 with a thermal power of just 58.3MW and
1When a nucleus of a 235U target captures a neutron, it splits into two or more lighter nuclei, releasing
kinetic energy of typically 200MeV per fission, gamma radiation, and free neutrons. In a nuclear chain
reaction, a portion of these neutrons is later captured by other fissile atoms and triggers further fission
events, which in turn release more neutrons.
81
82 CHAPTER 4. MEASUREMENTS AT THE ILL
Figure 4.1: A sketch of the experimental set-up of the aSPECT experiment at the PF1b of the
ILL. Unpolarized, cold neutrons (light green) are coming from the vertical liquid deu-
terium cold source (to the left of the scheme). Different equipment for neutron beam
preparation, like velocity selector, polarizer, spin flipper, or chopper, can be installed
inside the casemate. In our experiment, the neutrons are guided by an additional neu-
tron guide (black). The beam is shaped by the collimation system, comprising of B4C
(turquoise) and 6LiF (white) apertures. It flies through the aSPECT magnet (dark
green) to a beam stop. The stability of the neutron beam intensity was continuously
monitored with a 6Li neutron counter at the end of the beam stop. In between the
end of the guide H113 and our additional neutron guide a B4C shutter was installed,
which allows automated background measurements.
the most in∫tense continuous neutron flux in the world:
Φ = dvΦ(v) = 1.5× 1015 cm−2 s−1. (4.1)
Its core comprises a single highly enriched 235U fuel element cooled by heavy water. The
fission neutrons are very high-energy neutrons, with velocities of about 20000 km s−1.
They are slowed down by the heavy water to thermal neutrons, with velocities of about
2200m s−1. This is still much too fast for nuclear and particle physics experiments, where
the neutron decay rate is inversely proportional to the neutron’s velocity. Hence, the
thermal neutrons are further slowed down to the desired energy by one of two cold sources,
i.e., 25K liquid deuterium moderators near the reactor core. These cold neutrons have
velocities of about 800m s−1. The neutrons are extracted from inside the reactor by
neutron guides, which in turn distribute the neutrons to forty experimental areas (for
research in materials science, biology, physics, and chemistry; as well for the production
of radioisotopes for medical therapy and diagnosis), located up to 100m from the reactor.
4.1.1 The Neutron Beam Facility PF1b of the ILL
The cold (polarized) neutron beam facility PF1b is installed at the vertical liquid deu-
terium cold source of the ILL, at the end position of the very intense ballistic supermirror2
(m = 2) guide H113 [213]. The instrument layout is schematically shown in Fig. 4.1. The
neutron guide H113 ends in a casemate, right in front of the experimental zone. The
2In contrast to a conventional, single layer nickel coated neutron guide (m = 1), a neutron supermirror
consists of typically 100 double layers of nickel and titanium, of varying thickness. In this way, the critical
angle of total reflection is increased by a factor of two or more (m ≥ 2) and hence the transmission by up
to a factor of four.
4.1. EXPERIMENTAL SET-UP AT THE ILL 83
different equipment for neutron beam preparation, like focusing beam guides, neutron ve-
locity selector, supermirror polarizer, spin flipper, or chopper, can be installed inside the
casemate. T∫he guide H113 offers a thermal equivalent neutron flux3 of [214]
v
Φ = dvΦ(v) 0 = 1.8× 1010 cm−2 s−1c (4.2)
v
over a cross-section of 60 × 200mm2 [215], where v0 = 2200ms−1 is the most probable
velocity of a Maxwellian (thermal) spectrum at 300K. We note that, in most nuclear and
particle physics experiments, the capture flux Φc is more interesting than the particle flux
Φ, as the detection efficiency of most neutron detectors is inversely proportional to the
neutron’s velocity. For cold neutrons, with v < v0, the capture flux is higher than the
particle flux.
Further characteristics of the instrument can be found in Ref. [215].
4.1.2 The Beam Line
Figure 4.1 shows a scheme of the experimental set-up of the aSPECT experiment at the
instrument PF1b. Unpolarized, cold neutrons are guided (and shaped) by the beam line
from the end of the neutron guide H113 to the spectrometer. The beam line comprises of
• an (additional) sintered boron carbide4 (B4C) shutter, which allows automated back-
ground measurements,
• an additional neutron guide with a cross-section of 50×116mm2, as aSPECT makes
no use of additional equipment like velocity selector, polarizer, or chopper,
• three apertures from B4C in the beam tube before the spectrometer,
• five apertures from isotopically enriched 6LiF inside the spectrometer (see also Fn. 4),
three before and two behind the decay volume (DV),
• a B4C beam stop, with a small hole in the middle to allow some neutrons to pass
through to
• a 6Li neutron counter (see also Sec. 4.2.1) at the end of the beam stop, to continu-
ously monitor the stability of the neutron beam intensity, and
• boron loaded rubber, in and outside the beam tube, followed by 5 to 10 cm of lead,
for radiation protection.
Magnesium cast alloy (MgAl3Zn1) windows, with a thickness of 250µm, were installed
at both the entrance window and the exit window of the aSPECT magnet, in order to
separate the rough vacua of the collimation system (≈ 10−2 mbar) from the ultra-high
vacuum (UHV) inside the spectrometer (< 8× 10−9 mbar; see also Sec. 3.4.3). Figure 4.2
shows photographs of the experimental set-up at PF1b, with a view to both the entrance
and the exit side of the aSPECT spectrometer.
3Also called the capture flux.
4Cold neutrons are easily absorbed by neutron capture in materials with a high neutron absorption
cross-section. Most materials release primary gamma radiation, while others trigger secondary reactions,
which in turn could produce fast neutrons. Inside the aSPECT spectrometer the use of isotopically
enriched 6LiF is preferable, as the production of gamma rays is suppressed by a factor of 10−4 compared
to boron (10B) and as most materials containing 10B are not suitable for ultra high vacuum.
84 CHAPTER 4. MEASUREMENTS AT THE ILL
(a) (b)
Figure 4.2: Photographs of the experimental set-up of the aSPECT experiment at PF1b of the
ILL. Figure (a) shows a view to the entrance side of the spectrometer, while Fig. (b)
shows a view to its exit side. Unpolarized, cold neutrons (green) pass through the decay
volume where only a few neutrons decay. The decay protons (red) are guided by the
strong magnetic field (light blue) towards the analyzing plane (located between the
Helmholtz coils c14 and c15 and not shown here) and subsequently the proton detector
(white). aSPECT measures the proton recoil spectrum (see Fig. 2.3) by counting all
decay protons that overcome an electrostatic barrier, applied in the analyzing plane.
The photographs also show the magnetic field return, which serves to reduce the
exterior magnetic field in order not to disturb neighboring experiments (for details see
Chap. A). We note that the photo (a) was taken before the additional Helmholtz coils
c14 and c15 were added (see also Sec. 3.2.2).
In the first half (November/December 2007) of our beam time at the ILL, the beam
line with its collimation system and radiation protection was set-up, the superconducting
magnet with its magnetic shielding (for details see Chap. A) and electrode system (de-
scribed in Sec. 3.2.1) were installed, and the new SDD was tested as a proton detector
4.1. EXPERIMENTAL SET-UP AT THE ILL 85
(see Sec. 3.3.1 for details). The main phase of data taking took place in the second half
(April/May 2008) of the beam time (discussed in Sec. 4.2.5 and Chap. 5). Further details
on the beam line, the collimation system, and the alignment of the aSPECT spectrometer
with respect to the neutron beam can be found in Refs. [34, 52].
The neutron beam was shaped by the additional neutron guide to a cross-section of
48 × 72.5mm2, measured by copper foil activation (for details see the following section
and Ref. [34]) at the entrance of the aSPECT magnet. The thermal equivalent neutron
flux was measured by gold foil activation at the exit flange to be
Φc = 6× 109 cm−2 s−1 (4.3)
over a cross-section of 54 × 90mm2. The beam profile was measured by copper foil acti-
vation.
4.1.3 Neutron Beam Profiles
As discussed earlier in Sec. 3.4.6, the neutron beam profile has to be sufficiently homoge-
neous or its shape has to be known precisely. To minimize the edge effect, the neutron
apertures were optimized, by means of Monte Carlo (MC) simulations, to obtain a beam
profile as uniform as possible. To verify our simulations [34], the neutron beam profile was
measured by copper foil activation both in front of the entrance window and behind the exit
window of the aSPECT spectrometer. For this purpose, thin foils (thickness ≈ 150µm)
of natural copper (composed of 69.15(15)% of 63Cu and 30.85(15)% of 65Cu) were irra-
diated by neutrons for about 80min. Neutrons are captured by the two copper isotopes,
63Cu and 65Cu, and form the heavier isotopes 64Cu and 66Cu. The new isotopes are beta
emitters which decay into the nickel and zinc isotopes 64Ni or 64Zn and 66Zn, respectively.
Here, the half-lifes are τ = 12.700(2) h for 64Cu [216] and τ = 5.120(14)min for 66Cu
[217]. After a decay time of another 60 to 120min, the two-dimensional neutron beam
profiles were read out (see also [218]). For this purpose, the decay electrons, of almost
only 64Cu, were detected by a silicon PIN diode5. The silicon PIN diode was mounted to
a robotic arm above the irradiated copper foils, which allowed to scan the copper foils in
a grid pattern (for details see [34]). We mention that the overall spatial accuracy of our
measured beam profiles is 1mm, due to the uncertainties of the positioning both in the
beam line and in the scanner.
Figure 4.3 shows the measured beam profiles in comparison with the simulated ones.
After adaption of the input parameters for the MC simulation to the measured beam pro-
files a reasonable agreement was found. However, for the calculation of various systematic
corrections, we need the neutron beam profile in the DV of aSPECT. At the time of our
beam profile measurements, a measurement of the neutron beam profile directly in the
DV was not feasible. Hence, we had to go for the simulated beam profiles in the DV
instead. Therefore, the input parameters for the MC simulation were adjusted until the
simulated beam profiles both in front and behind the spectrometer were consistent with
the measured ones. We mention that, for the investigation of systematic effects, a cut at
x = 0 is sufficient as the neutron beam profile varies only marginally over the depth, x, of
the flux tube.
For systematic investigations of the edge effect (see Secs. 3.4.6 and 5.5.6 for details),
measurements with two different reduced widths of the neutron beam were carried out.
5Normally, the neutron beam profile of an irradiated copper foil is read out after the transfer to an image
plate. Unfortunately, the image plate scanner was damaged at the time of our beam profile measurements.
86 CHAPTER 4. MEASUREMENTS AT THE ILL
(a) (b)
(c) (d)
Figure 4.3: Neutron beam profiles measured by copper foil activation (top) in front of the entrance
window and (bottom) behind the exit window of the aSPECT spectrometer. (a) and
(c) show the two-dimensional, measured beam profiles; whereas (b) and (d) show
horizontal cuts through the beam profiles (a) and (c), respectively, in comparison with
the simulated beam profiles, at z = 0. Even though a reasonable agreement between
the measured and the simulated beam profiles was found, the neutron beam profile in
the DV was determined and subsequently smoothed both by bilinear interpolation of
the measured beam profiles (a) and (c) (discussed also in Sec. 5.5.6). Error bars show
statistical errors only.
For this purpose, an additional aperture was installed in front of the aperture P2, cf.
Fig. 4.1. Figure 4.4 shows the neutron beam profiles for the 20mm and the 5mm wide
aperture, respectively, both measured behind the exit window of aSPECT.
But even though a reasonable agreement between the measured and the simulated
beam profiles was found, the poor quality of the simulated beam profiles is not ade-
quate as input data for further MC simulations of, e.g., the edge effect. Hence, the two-
dimensional, simulated beam profiles were smoothed by bilinear interpolation (described,
e.g., in Ref. [90]). In the case of Fig. 4.3d, the agreement between the measured and the
simulated beam profile is so poor that we had to determine the neutron beam profile in the
DV by prior bilinear interpolation of the measured beam profiles in front of the entrance
window and behind the exit window of the aSPECT magnet (discussed in Sec. 5.5.6).
4.1. EXPERIMENTAL SET-UP AT THE ILL 87
(a) (b)
(c) (d)
Figure 4.4: Neutron beam profiles measured by copper foil activation (top) for the 20mm and
(bottom) the 5mm wide aperture. (a) and (c) show the two-dimensional, measured
beam profiles; whereas (b) and (d) show horizontal cuts through the beam profiles
(a) and (c), respectively, at z = 0. For comparison, Fig. (b) also shows the simulated
beam profile. Even though a reasonable agreement between the measured and the
simulated beam profiles was found, the neutron beam profile in the DV was smoothed
by bilinear interpolation of the simulated beam profiles. Error bars show statistical
errors only.
We note that the neutron beam profiles shown in Figs. 4.3a to 4.4d deviate from those
presented in Refs. [34, 52, 53]. Owing to a coordinate system rotated by -90° in the x-
y-plane the beam profiles were presented inversely. Therefore, the relationship x = −y
applies to the width of the neutron beam profile.
Further details on the simulations and measurements of the neutron beam profile can
be found in Ref. [34].
4.1.4 Magnetic Field Profiles
As discussed earlier in Sec. 3.1.3, the magnetic field ratio rB = BA/B0 has to be known
with a relative accuracy of ∆r −4B/rB = 1× 10 , in order to keep systematic uncertainties
88 CHAPTER 4. MEASUREMENTS AT THE ILL
(a)
(b) (c)
Figure 4.5: The magnetic field profiles (a) along the z-axis, (b) in the decay volume (DV), and (c)
in the analyzing plane (AP) of the aSPECT spectrometer. The black diamonds are
measured on-axis, whereas the colored diamonds are measured off-axis at the edge of
the flux tube. For comparison, the blue lines show the (adapted) simulated magnetic
field profiles. See the text for details. The magnetic field profiles were measured (top)
at Imain = 70A and I3 = I5 = I12 = . . . = I15 = 0 and (bottom) at Imain = 70A,
I3 = 35A, I5 = 15A, I12 = −I13 = 25.5A, and I14 = I15 = 0. The error bars
represent the instability of the measurements, cf. Ref. [53].
in the neutrino-electron correlation coefficient a below ∆a/a = 0.1%. Therefore, we have
to know the magnetic fields in the analyzing plane (AP) and in the DV with a relative
accuracy of ∆BA/BA = 1×10−4 and ∆B0/B0 = 1×10−4, respectively (see also Sec. 3.4.1).
To verify our calculations and to determine the magnetic field ratio rB, the magnetic field
was measured at the beam position, before and after6 the beam time. The magnetic field
was measured with a high accuracy, fully temperature compensated Hall probe7, operated
at room temperature. So in order to measure the magnetic field, the electrode system
had to be removed and an inverted non-magnetic dewar had to be installed instead. The
6Shortly before the beam time, i.e., after the first magnetic field measurements at the beam position,
the superconducting aSPECT magnet quenched, due to an interruption in the cooling water supply.
To investigate the possible influence of the quench on the magnetic field ratio rB or the magnetic field
profile, the magnetic field was remeasured after the beam time. No difference was found between the
measurements before and after the beam time [53].
7A Group3 Technologies miniature hall probe MPT-141 [219].
4.1. EXPERIMENTAL SET-UP AT THE ILL 89
Table 4.1: The range of the magnetic field in the DV and the AP [53]. The inhomogeneity is out
of our tolerance of 1× 10−4. See Sec. 3.4.1, Eq. (3.46), and the text for details.
Direction (B0,max −B0,min) /B0 (BA,max −BA,min) /BA
longitudinal 7.4(4)× 10−4 0.3(5)× 10−4
radial 4(3)× 10−4 3(3)× 10−4
results of the magnetic field measurements are shown in Fig. 4.5, in comparison with the
(adapted) simulated magnetic field profiles.
Figure 4.5a shows the magnetic field profile along the z-axis for only Imain = 70A. The
difference between the measured and the simulated magnetic field profile at the height of
the HV electrodes e16A, e16B, and e17 is due to the fact that the calibration of the Hall
probe is valid only up to 2.2T. Figures 4.5b and 4.5c show a zoom to the DV and the AP,
respectively. As mentioned earlier in Sec. 3.2.2, the correction coils c3, c5, c12, and c13
were necessary to correct for small deviations of the magnetic field from its design shape
(see [168] for details). After adaption of the input parameters8 for the magnetic field
calculations to the measured magnetic field profiles a reasonable agreement was found.
We note that the positions of the local maxima both in the DV and the AP are consistent
with our calculations at the level of 0.5 cm. In particular, the positions of the local maxima
of the magnetic field (simulated at z = 132 cm and measured at z ≈ 131.5 cm [53]) and the
electrostatic potential (simulated at z = 132 cm) in the AP differ only by about 0.5 cm,
what automatically is compensated by the shrinkage of the electrode system by cooling
down. Consequently, the adapted parameters, i.e., the fitted coil currents, served as input
data for further MC simulations of, e.g., the edge effect.
The experimental data points off-axis were measured at the edge of the flux tube
that connects the DV and the central detector pad9. For Imain = 70A (Imain = 30A), the
diagonal of this flux tube is d ≈ 2.6 cm (d ≈ 3.6 cm) in the DV and d ≈ 5.8 cm (d ≈ 7.9 cm)
in the AP, taking account of the round edges of the detector pads (see Fig. 3.18). The
measured magnetic field shows axially symmetry10. The range of the magnetic field values
in the DV and in the AP
B0,max −B0,min BA,max −Band A,min (4.4)
B0 BA
is listed in Table 4.1 (see also [53]). From these magnetic field values, we obtain an average
magnetic field ratio of11 (cf. also Sec. 3.4.1 and Eq. (3.46))
rB = 0.20299(6), (4.5)
with a relative error of 3×10−4 dominated by the accuracy of the Hall probe. In particular,
the implemented calibration of the Hall probe is out of date. This yields a systematic error
8At the time of the analysis of several different systematic effects, the final analysis of the magnetic
field measurements had not been completed. Hence, only the currents and not also the positions of the
superconducting coils were fitted to the measured magnetic field profiles.
9For Imain = 70A, this flux tube also covers half of both the left and right detector pads.
10We suspect that the slightly different shape of the magnetic field in the AP for ϕ = 90° represents the
instability of the Hall probe [53].
11The error includes a scale factor of about 2/3, as generally we have to average over the on-axis and
two off-axis measurements.
90 CHAPTER 4. MEASUREMENTS AT THE ILL
in the angular correlation coefficient a of
∆a/a = ±0.26(10)% (4.6)
Further details on the measurements of the magnetic field profile, the determination,
stability, and reproducibility of the magnetic field ratio rB, and the development of and
first tests with an online nuclear magnetic resonance system can be found in Ref. [53].
4.2 Data Acquisition
The aSPECT data acquisition system (DAQ) was introduced in Sec. 3.3. In the following,
we briefly discuss further details on the DAQ during our latest beam time at the ILL.
4.2.1 Neutron Beam Monitor
For a high-precision measurement of the neutrino-electron angular correlation coefficient
a, for each measurement file the measured count rates have to be normalized to the number
of (unpolarized, cold) neutrons that pass through the DV during the measurement time
t (for details see Sec. 4.2.4). In the course of a beam time, this number may fluctuate
due to changing conditions of the instrument set-up or the cold source (see Sec. 4.1), or a
changing reactor power. The 6Li neutron counter (introduced in Sec. 4.1.2) was therefore
intended to continuously monitor the stability of the neutron beam intensity. The neutron
counter showed fluctuations at a high level that was neither correlated to fluctuations in
the reactor power nor in the measured proton count rates. It therefore makes little sense
to normalize the measured proton count rates to the measured neutron count rates.
Details on the neutron counter performance can be found in Ref. [52].
4.2.2 Monitoring of the Barrier Potential
The voltage applied to the AP electrode e14 is provided by a custom-designed, high accu-
racy power supply12 and monitored by a calibrated, precise multimeter13. The accuracy
of the voltage settings is limited by the calibration of the multimeter to 4mV. However,
the stability and the reproducibility of the barrier voltage were both found at the level
of 1mV, i.e., much better than the calibration of the multimeter. This considerably sim-
plifies the extraction of the neutrino-electron correlation coefficient a from the measured
proton spectra (discussed in Sec. 5.2), as in the fit of the integral proton spectrum, for each
barrier voltage, the average count rate can be used (see Secs. 5.2.2 and 5.2.3 for details).
In addition, for each barrier voltage, the voltage selected at the power supply was used.
But the true value of the voltage UA differs from the selected one by up to 11mV, as can
be seen from Table 4.2. This introduces a shift in a of ∆a/a = −0.08(11)% before and
∆a/a = −0.11(11)% after correction for the multimeter’s calibration. After additional
correction for the penetration of electric field (see page 58 under “The Analyzing Plane”),
this method introduces a shift in the angular correlation coefficient a of
∆a/a = −0.09(11)% (4.7)
4.2. DATA ACQUISITION 91
Table 4.2: Typical voltage settings of the analyzing plane (AP) electrode e14. The four columns
contain, from left to right, the voltage UA selected at the power supply (see Fn. 12), its
average value (avg) monitored by the precise multimeter (see Fn. 13), its average value
after correction for the multimeter’s calibration, and the corresponding electrostatic
potential in the center of the AP.
UA avg UA avg UA Potential in
selected monitored corrected center of AP
[V] [V] [V] [V]
50 49.995 49.995 49.995
250 249.997 249.998 249.997
400 400.008 400.010 400.009
500 500.008 500.011 500.009
600 600.008 600.011 600.009
Details on the ramping and the stability of the barrier voltage can be found in Ref. [52].
4.2.3 Energy Calibration of the Proton Detector
As mentioned earlier in Sec. 3.3, for each measurement file the decoding program creates
a pulse height histogram (in ADC channels), cf., e.g., Fig. 3.19b. Here, the pulse height is
proportional to the particle energy deposited in the active layer of the SDD. To convert the
pulse height into particle energy, our SDD was calibrated with 133Ba before installation
in the aSPECT spectrometer.
133Ba disintegrates by electron capture to 133Cs. The decay scheme is rather compli-
cate, with several gamma transition, X-ray emission, and Auger escape lines [221, 222].
Figure 4.6 shows a calibration spectrum, recorded at room temperature, together with
the fit to the Kα Auger escape, the Kα, and the Kβ line. From the mean values of the
Gaussian distributions, the following linear relationship between pulse height and particle
energy at ro(om temperature was derived: )
ph = 65.8 (3) keV−1 · Edet + (−4.4± 8) ADC channels, (4.8)
where ph is the detected pulse height and Edet is the energy detected by the SDD.
We note that the conversion factor of 65.8(3) keV−1 in Eq. (4.8) differs slightly from the
result of our proton simulations of 63 keV−1 [52]. We are convinced that this difference
is due to the different temperature of both measurements, as the proton spectra were
recorded with the cooled SDD (see also [51]).
4.2.4 Measurement Sequence
As described earlier in Sec. 3.1.3, the neutrino-electron correlation coefficient a is derived
from the proton spectrum shape, which, in turn, is measured by counting all decay protons
that overcome the electrostatic barrier, UA. During data taking, measurements with seven
12A FUG Elektronik high voltage power supply HCN 0,8M-800.
13Currently an Agilent Technologies 3458A digital multimeter.
92 CHAPTER 4. MEASUREMENTS AT THE ILL
Figure 4.6: A calibration spectrum of our proton detector with 133Ba, recorded at room temper-
ature. The black line is the fit to the Kα Auger escape, the Kα, and the Kβ line. At
room temperature, the resolution of the detector is not sufficient to separate the single
Kα and Kβ lines. Thus, the Kα Auger escape, the Kα, and the Kβ line were fitted with
three superimposed Gaussian distributions with mean values EAuger = 29.11(10) keV
[52] (see also [220]), EKα = 30.85(10) keV, and EK = 35.1(2) keV [221]. Error barsβ
show statistical errors only. Figure taken from Ref. [52].
different barrier voltages UA were taken. In fact, measurements with only three different
barrier voltages are required to
• quantify the background and
• determine the two fit parameters of the proton count rate Eq. (3.32), i.e., the full
proton decay rate N0 and the angular correlation coefficient a.
Measurements with UA = 50V were used to determine the full proton decay rate N0 (see
Secs. 3.1.3 and also 3.4.3), whereas measurements with UA = 780V served to quantify
the background (see Secs. 3.1 and also 3.4.4). A further measurement with UA ≈ 400V
would provide the best statistical sensitivity to the angular correlation coefficient a [46].
However, to obtain a more precise knowledge of the proton spectrum shape and detailed
information on the background, measurements with UA = 0, 50, 250, 400, 500, 600, and
780V were carried out. In particular, measurements with UA = 0V were used to remove
trapped particles. Typically, data taking cycles started with a measurement at UA = 0V,
followed by a measurement with UA = 50V, then a measurement with an intermediate
barrier voltage UA = 250, 400, 500, or 600V, and finally a background measurement
with UA = 780V. After the background measurement, this cycle was repeated with a
different intermediate barrier voltage. Every fifth cycle, the order of the measurements
with UA = 50V and an intermediate barrier voltage were interchanged. For automated
background measurements, each measurement, in turn, was subdivided into five neutron
shutter statuses, i.e., (1) the neutron shutter was closed for 10 s, then (2) opening during
1.1 s,14 (3) open for t s of measurement, then (4) closing during 0.6 s (see also Fn. 14), and
finally (5) closed for another 10 s. Here, the measurement time t was different for every
14Please note that in Ref. [34] the times for opening and closing the neutron shutter are misstated.
4.2. DATA ACQUISITION 93
barrier voltage, in order to optimize the measurement sequence for statistical sensitivity
(see also [46]). Measurements with opened neutron shutter took t = 10 s at UA = 0V,
t = 40 s at UA = 50V and 780V, and t = 120 s at UA = 250, 400, 500, and 600V. In
this way, measurements with UA = 50V lasted 4/3 times as long as measurements with
UA = 250, 400, 500, or 600V.
4.2.5 Investigation of Systematic Effects
Systematic effects, discussed thoroughly in Refs. [46, 47] and Sec. 3.4, were investigated
experimentally. Here, we only summarize the essential measurements:
Additional neutron shutter: For automated background measurements, an additional
neutron shutter was installed inside the casemate (see Sec. 4.1.2 for details).
Different settings of the lower dipole electrode: The lower dipole electrode e8 is
used to sweep out all decay protons that cannot pass the potential barrier and would
otherwise be trapped between the electrostatic mirror and the AP. To investigate
particle trapping between the electrostatic mirror and the AP (see also Sec. 3.4.3
under “Elastic Scattering”), the potential barrier was set to UA = 780V, so that no
protons should be able to overcome the potential barrier, and the drift potential was
reduced in several steps from U8R|U8L = −50V|−1000V to 0V|−2.5V (discussed in
Sec. 5.3.2).
Different settings of the upper dipole electrode: As discussed earlier on page 59
under “The Dipole Electrodes”, the upper dipole electrode e16 is used to align the
neutron beam on the proton detector. Assuming a neutron beam profile symmetrical
around its maximum (cf. Sec. 4.1.3), the edge effect is minimal, when the center of
the neutron beam is aligned on the center of the detector. To determine the position
of the high voltage electrode e17, and hence of the proton detector, relative to the
rest of the electrode system and the magnetic field, and to study the edge effect,
measurements at several different settings of the upper dipole electrode e16 were
carried out. For details see Sec. 5.5.6.
Different ratio of the magnetic fields: The Helmholtz coils c14 and c15 were used to
investigate the adiabatic transmission function Eq. (3.30). For I14 = I15 = 50A,
the magnetic field BA in the AP and hence the magnetic field ratio rB = BA/B0
change by about +1%. This allows a quick and statistically significant test of the
transmission function (discussed in Sec. 5.5.4).
Different height of the main magnetic field: During our beam time two different
magnetic field values B0 = 0.933T (Imain = 30A) and B0 = 2.177T (Imain = 70A)
were used (see Sec. 5.5.5), in order to study the influence of the non-adiabatic pro-
ton motion on the neutrino-electron correlation coefficient a (see also Secs. 3.4.2
and 5.4.4). We mention that with decreasing coil current Imain also the radii of gy-
ration Eq. (3.3) increase and hence the edge effect changes (for details see Sec. 5.5.6).
Electrostatic mirror switched off: Due to the magnetic mirror effect (discussed on
page 46 under “Magnetic Mirror Effect”) about 1% of the decay protons, emitted in
the negative z-direction, are reflected by the local magnetic field maximum below the
neutron beam (see also Fig. 4.5b). To investigate the alignment between magnetic
94 CHAPTER 4. MEASUREMENTS AT THE ILL
field and neutron beam, the proton count rates with and without electrostatic mirror
were compared (see Sec. 5.5.2 for details).
Electric field gradient in the DV: As discussed in greater detail later in Sec. 6.4.3,
work function inhomogeneities in the DV will lead to unexpected proton reflections
from the DV. Approaches for this effect are presented in Sec. 6.4.4. Due to a lack of
time, only different “Inverse electric field gradient[s]” were investigated during our
beam time. For this a potential of a few volts was applied to the bottom and top
cylinders e3 and e6 of the DV electrode, with opposite sign (for details see Sec. 5.3.3).
Second analyzing plane between DV and AP: To study possible background con-
tributions of electrons, the lower dipole electrode e8 was used as a second analyzing
plane (see also Sec. 3.4.4 under “Uncorrelated background events”). For this purpose,
a positive potential of up to +1000V was applied to sides R and L of the dipole
electrode (see Sec. 5.3.1).
Different neutron beam profiles: As mentioned earlier in Sec. 4.1.3, measurements
with three different widths of the neutron beam were carried out, in order to inves-
tigate the edge effect (see Secs. 3.4.6 and 5.5.6 for details).
Different trigger settings: To study the efficiency of our detection system, the windows
w1 and w2, the delay, and the threshold (introduced on page 69 under “Digital
Electronics and the Trigger Algorithm”) were varied over a wide range. It was found
that the measured count rate is independent of the trigger settings, provided that
the window w1 is less than or equal 24 time bins (0.8µs). For details the reader is
referred to Ref. [52].
Different post-acceleration voltages: During our beam time post-acceleration volt-
ages from -10 to -15 kV were used, in order to investigate the detection efficiency of
our proton detector. The influence of backscattering of decay protons on the angular
correlation coefficient a was already discussed in Sec. 3.4.7 (see also [52]).
Chapter 5
Data Analysis
In the previous chapter, we discussed the measurements during our latest beam time at
the Institut Laue-Langevin (ILL) in Grenoble, France. In this chapter, in turn, we present
the detailed analysis of the measured data.
The raw data analysis was mostly carried out by M. Simson [52] and M. Borg [34]. To
avoid a biased analysis, a blind data analysis was performed. Therefore the values of a
extracted from the measured count rates were transformed by means of a linear function:
acheat = m · a+ b, (5.1)
where the parameters m and b were selected so that the influence on the error bar of a
is significantly small, i.e., b/m  1. Both parameters were only known to me and two
other persons not directly involved in the analysis. This made it impossible to push the
extracted value of a towards or away from previously measured values, neither deliberately
nor unconsciously. After the analysis was completed the figures were revealed to be1:
m = 1.2477 and
b = −0.0008. (5.2)
Then the values of a were corrected according to Eq. (5.1).
The main emphasis of this thesis lies on the study of systematic effects. Exhaustive
Monte Carlo (MC) simulations were performed to compute several systematic corrections.
For details on the raw data analysis, the reader is referred to the theses of M. Simson [52]
and M. Borg [34].
We note that the pictures and examples in this chapter stem from measurements at
an acceleration potential of -15 kV, unless otherwise stated.
5.1 Data Fitting
As described earlier in Sec. 3.3.2, for every trigger an event as shown in Fig. 5.1 is stored.
To extract the proton spectrum for one measurement run the pulse height of each event
has to be determined. The simplest way to calculate the pulse height of an event is to
determine its maximum and to subtract a fixed baseline from its maximum. The result of
this method is denoted as hnf . However, this method is insensitive to possible fluctuations
1Please note that in Ref. [52] the parameter b is misstated with positive sign.
95
96 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.1: Typical neutron decay events. (a) A proton event with its resulting fit (red line):
x0 = 20.0(1), y0 = 1489(6), and A = 29040(455). The corresponding pulse height
is hfit = 554.0 compared to hnf = 549.8 from the “maximum−baseline” method.
(b) An high energy electron or gamma event. The pulse is cut off by the maxi-
mum amplification of the electronics. In our analysis, such events are not fitted as the
fitting routine gives nonsensical results (red line) due to the different pulse shape. See
the text for details.
of the baseline. To avoid this problem, the baseline can be calculated separately for
each event from the first 15 ADC bins. Nevertheless, the maximum of the pulse could
be incorrectly determined due to, e.g., spikes from the electronic noise. Therefore, we
concentrated on fitting the events.
5.1.1 The Fit Function
The pulse shape of a normal event can be described as a product of two exponential
functions withdifferent time constants [197]: y0 ( ) , x ≤ x0yfit(x) = −x−x p0 −x−x0 . (5.3)y t t0 +A 1− e 1 e 2 , x > x0
Here, y0 is the baseline, x0 is the start point, t1 and t2 are the rise and decay time
of the pulse, respectively, and A and p describe the height and the shape of the pulse,
respectively. Table 5.1 presents the exact limits of the fit parameters2. The quantities
t1, t2, and p are defined by the detector electronics and are therefore frozen in the fitting
2Please note that the fixed values of the parameters t1 and p deviate from those given in Refs. [34, 52].
Owing to a misplaced bracket in the fitting routine the events were fitted to a function slightly different
from Eq. (5.3):
 y0 ( ( ) ) , x ≤ x0′ 1y(x) =  − x−x0 p′ − x−x0 . (5.4)y0 +A 1− e t1 e t2 , x > x0
Therefore, the relationship t = t′ /p′1 1 applies to the fit parameters.
5.1. DATA FITTING 97
Table 5.1: Limits of the used fit parameters. The time constants t1 and t2 and the parameter p
are defined by the electronics and thus frozen in the fitting routine.
Parameter Unit Lower limit Upper limit
y0 ADC channels 800 2000
x0 ADC bins (×50 ns) 13 23.5
t1 ADC bins (×50 ns) 196.675 (frozen)
t2 ADC bins (×50 ns) 10.47 (frozen)
p ÷ 1 (frozen)
A ADC channels 1 2 · 105
Table 5.2: Overview of the different fit states. A reduced chi-squared χ2/ (hfit · ndof) < 0.00741
indicates a good fit whereas χ2/ (hfit · ndof) ≥ 0.00741 indicates a poor fit to the model.
Events with a bad χ2 are sub-classified according to their pulse height, as electronic
noise events have a hfit < 80. High energy events are not fitted as they are cut off by
the maximum amplification of the electronics. Events where the fit did not converge
or with a bad χ2 are refitted with adapted fitting routines. See the text for details.
Status Description
0 Normal events with good χ2
1 Events with bad χ2 and hfit ≥ 80
2 Events with bad χ2 and hfit < 80
3 High energy events
4 Fit failed
5 Refitted status 1 events
6 Refitted status 2 events
7 Events with different pulse shape
routine. The start point x0 depends on the settings of the trigger algorithm and slightly
on the pulse height. Thus it is not fully fixed in the fitting routine.
To obtain the pulse height of an event, function Eq. (5.3) is fitted to the pulse. The
fit is limited to ADC bins 10 to 40, to significantly reduce the computing power and,
consequently, the fitting time. The maximum point of the fit depends on the fit parameters
as follows:
xmax = x0 + t1 (ln (t1 + pt2)− ln t1) ≈ x0 + 10.2 ADC bins. (5.5)
By substituting xmax into Eq. ((5.3) we ge)t th(e fitted p)ulse height, termed as hfit:p t1/t2
hfit = yfit(xmax)−
pt2 t1
y0 = A ≈ A · 0.019 ADC ch.−1.(5.6)
t1 + pt2 t1 + pt2
5.1.2 Fit States
According to the result of their fit the events are classified into different categories. In
Table 5.2 we list the different fit states.
98 CHAPTER 5. DATA ANALYSIS
Since high energy electrons and gammas are cut off by the limited maximum amplifi-
cation of the electronics, the exact pulse height of these events cannot be reproduced by
the fit. Thus only events with ADC values below 3500 before ADC bin 30 are fitted.
Events where the fit did converge are classified according to the reduced chi-squared.
Since electronic noise events have a hfit < 80, events with a bad χ2/ (hfit · ndof) ≥ 0.00741
are additionally classified according to their pulse height, where ndof is the number of
degrees of freedom. To achieve the best possible separation between good and bad events
the limits were chosen manually3. Two examples for fit states 0 and 3 are shown in Fig. 5.1.
Events where the fit did not converge are refitted with the improved fitting routine of
ROOT. If the fit still does not converge, the pulse is rebinned, to even out short spikes in
the electronic noise, and fitted again. In this way, practically all events can be fitted.
5.1.3 Refitted Events
Unfortunately, not all events are sorted correctly by the standard fitting routine. Espe-
cially in two cases valid proton events are sorted incorrectly into fit status 1 or 2:
Pile-up: When a second proton is detected immediately after the triggering one (before
ADC bin 40) the fit does not work as the rising edge of the second proton changes
the trailing edge of the triggering one. Top Fig. 5.2a shows a pile-up event.
Proton after electron: When a proton is triggered shortly after an high energy electron
its baseline is changed by the exponentially decaying edge of the electron, which in
turn is longer than one event window. Top Fig. 5.2b shows such an event.
To identify and separate the incorrectly sorted events from the correctly sorted ones
the derivative of the event shape is calculated. Two examples are shown in the bottom
Fig. 5.2. A steep rise in the signal corresponds to a significant peak in its derivative.
From the number and position of those peaks the two above mentioned cases can be
distinguished:
• Pile-up events have two peaks in their derivative, the first one around ADC bin 23
and the second one between ADC bin 25 and 40. Such events are refitted with the
standard fit function Eq. (5.3), but the fit is limited to ADC bin 5 to 1σ before
the second Gaussian peak. This method works well down to time differences of the
two proton events of about 0.25µs [52]. A comparison of the standard fit with the
adapted fit is shown in Fig. 5.2a.
• Protons shortly after high energy electrons have one single peak before ADC bin
13.5 in their derivative, as they are triggered later than normal events. Those events
are refitted with an adapted fit function:
yPaE(x) = y (x) + ep1x+x1fit − 1, (5.7)
where yfit is the standard fit function Eq. (5.3). The quantities p1 and x1 describe
the exponentially decaying edge of the electron. To improve the knowledge of the
changing baseline the fit is limited to ADC bin 1 to 35. We mention that the pulse
height of such events is also given by Eq. (5.6). A comparison of the standard fit
with the adapted fit is shown in Fig. 5.2b.
3Usually, a reduced chi-squared near 1.0 means that within the error bars one has a good fit to the
model. In our case, this rule of thumb is not valid as we assigned an error bar of 20 to each ADC value.
5.2. EXTRACTION OF A FROM THE PROTON SPECTRA 99
(a) hfit = 584.8 compared to hnf = 906.2 (b) hfit = 151.5 compared to hnf = 115.9
Figure 5.2: Valid proton events which were sorted incorrectly into fit status 1. Figure (a) shows a
pile-up event, whereas Fig. (b) shows a proton shortly after an high energy electron.
Top: The results of the standard fit and the adapted fit are shown in red and green,
respectively. According to the result of the adapted fit the events are assigned fit status
5. Bottom: To separate these events from the correctly sorted ones the derivative of
the event shape is calculated. A steep rise in the signal corresponds to a significant
peak in its derivative. From the number and position of those peaks the two cases (a)
and (b) can be distinguished. The positions of possible peaks are marked with red
triangles. The Gaussian fits to the relevant peak candidates are shown in blue.
Some valid proton events, which were incorrectly sorted into fit status 1, do not fall into
the above mentioned two categories. These events have a slightly different pulse shape:
The rising edge is not as steep as for the normal events and thus the maximum is shifted
to higher ADC bins. Such events are refitted with the standard fit function Eq. (5.3), but
with different fit limits. To focus on the falling edge the fit is limited to ADC bins 30 to
90. Additionally, the upper limit of the start point is extended to ADC bin 25.
The results of the fit are written to a ROOT tree for further analysis. More examples
and details on the structure of the fit tree can be found in Ref. [52].
Figure 5.3 shows a comparison of fitted with non-fitted pulse height spectra. Obviously
the data fitting improves the separation between the proton signal and the electronic noise.
5.2 Extraction of a from the Proton Spectra
From the fitted pulse height spectra (see Fig. 5.3) we now can extract the value of the
neutrino-electron correlation coefficient a. Before that, the count rate of each measure-
ment file has to be corrected for the dead time of the electronics. Then the background,
100 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.3: Typical pulse height spectra for one detector pad from 15_05_08/rampHV for -10 and
-15 kV acceleration potential in (a) linear and (b) logarithmic representation. Fitted
pulse height spectra are shown in red and green, non-fitted ones in blue and black.
With increasing acceleration potential the proton peak (right peak) is shifted to higher
ADC channels whereas the electronic noise (left peak) is almost not influenced. For
both potentials, the noise peak is shifted to lower ADC channels for the fitted spectra
whereas the proton peak remains almost the same. The fit evens out spikes from the
electronic noise on the proton events. Thus the proton peak is slightly shifted to lower
ADC channels for the fitted spectra. Error bars show statistical errors only.
measured at 780V barrier voltage, has to be subtracted. For each barrier voltage, the pro-
ton count rate has to be determined from the background subtracted pulse height spectra
by integration over the pulse height. To finally obtain the angular correlation coefficient
a, function Eq. (3.32) has to be fitted to the integral proton spectrum. In addition, several
different corrections have to be applied.
5.2.1 Dead Time Correction
As described earlier in Sec. 3.3.2, for every trigger an event with a length of normally
5µs is stored (see Fig. 5.1). During this time no other event can be registered, apart
from pile-up events. In the case of a pile-up event, the fitting routine ensures that only
the first proton is counted, as can be seen from Fig. 5.2a. Due to the processing time of
the electronics of 0.2µs, the next registered event has a minimum time difference to the
preceding one of 5.2µs. This defines a so-called non-extendable [223] dead time, τdead,
which causes a reduction of the proton count rate, Np(UA). The measured proton count
rate, Np,meas(UA), then depends on the total count rate
Ntotal(UA) = Np(UA) +Ne +Nnoise (5.8)
as follows:
Np,meas(UA) = Np(UA) (1−Ntotal(UA) · τdead) , (5.9)
where Ne = Ne(U17) and Nnoise are the electron and electronic noise count rates, respec-
tively. The actual proton count rate may be estimated by [223]:
Np,corr(
N
U p,meas
(UA)
A) = 1−N (5.10)total,meas(UA)·τdead
5.2. EXTRACTION OF A FROM THE PROTON SPECTRA 101
(a) (b)
Figure 5.4: Relative change of the angular correlation coefficient a for (a) different dead times of
the electronics and (b) different electronic noise count rates, before (red) and after
(green) correction for the dead time by Eq. (5.10). For better visibility, the y-axis
is broken at ∆a/a = 0.22. The dead time correction is the better, the shorter
the electronics dead time and the lower the electronic noise count rate are. In-
put data for the simulation (with Nachtmann’s formula Eq. (2.38)): N = 490 s−10 ,
UA = 50, 250, 400, 500, 600V, U17 = −15 kV, N −1noise = 500 s (a), τdead = 5.2µs (b),
and a = −0.103 [10]. The error bars, in the order of 0.05% after correction for the
dead time, represent the error by fitting Eq. (3.32) to the simulated proton count rates
only.
where
Ntotal,meas(UA) = Np,meas(UA) +Ne,meas +Nnoise,meas (5.11)
is the total measured count rate. Here, Ne,meas and Nnoise,meas are the measured electron
and electronic noise count rates, respectively. The measured count rates Ne,meas and
Nnoise,meas depend on the total count rate Ntotal(UA) in the same way as Eq. (5.9). Because
of the quite high count rate on one detector pad, the dead time correction has a significant
impact on the extracted value of the angular correlation coefficient a. Figure 5.4a shows
the relative change of the angular correlation coefficient a for different dead times of the
electronics, before and after correction for the dead time by Eq. (5.10). Assuming a full
proton decay rate of N0 = 490 s−1, an electronic noise count rate of N −1noise = 500 s , and
an electronics dead time of τdead = 5.2µs, the proton count rate loss at U17 = −15 kV
would be approximately 0.54%4 for UA = 50V down to 0.3% for UA = 600V. This would
correspond to a relative shift of the angular correlation coefficient a of +3.7% (see also
Fn. 4) before and of
∆a/a = +0.035(50)% (5.12)
after correction for the dead time. We mention that during our latest beam time at the
ILL the electronic noise count rate usually was around (250− 950) s−1; during the hottest
4Please note that the figures deviate from those given in Refs. [34, 52, 53], in which neither the electron
nor the electronic noise count rate were taken into account. In our example, the proton count rate loss
would be 2.5 s−1 at UA = 50V, two times higher than stated in Refs. [34, 52, 53].
102 CHAPTER 5. DATA ANALYSIS
days it reached in excess of 1450 s−1. Actually, one would not expect a shift of the angular
correlation coefficient a after correction for the dead time. At first order, Eq. (5.10) is a
good estimation for the actual proton count rate Np(UA), as follows from Eqs. (5.9) and
(5.10):
∆Np(UA) = Np(UA)−Np,corr(UA)
Np,meas(UA) − N= p,meas(UA)
1−Ntotal(UA) · τdead 1−Ntotal,meas(UA) · τdead
Np(UA) (1−Ntotal(UA) · τdead) [Ntotal(UA) · τdead −Ntotal,meas(UA) · τdead]=
(1−Ntotal(UA) · τdead) (1−Ntotal,meas(UA) · τdead)
Ntotal(UA) (Ntotal(UA) · τ= ( dead) · τdeadNp UA) (1−Ntotal,meas(UA) · τdead)
= Np[(UA)N2total(UA) · τ2dead ( )]
× 1 +Ntotal(UA) (1−Nt[otal(UA) · τdead) · τ 2dead +O(τdead)]
= N 2 2 2p(UA)N[(total(UA) · τdead 1 +Ntotal(UA) · τdead +O) τdead ( )]
= Np(U 2A) Np(UA) + 2Np(UA)Ne + 2N
2 2
p(UA)Nnoise · τdead +O τdead
( ) (5.13)
≈ N2p(UA) · O τ2dead .
But according to Eq. (5.13), the relative error due to the dead time correction,
∆Np(UA)/Np(UA), is proportional to both τ2dead and Np(UA) [Np(UA) + 2Ne + 2Nnoise].
Consequently, the correction for the dead time is the better, the shorter the electronics
dead time is, as shown in Fig. 5.4a. In addition, the dead time correction is the better,
the lower the electronic noise count rate is, as can be seen from Fig. 5.4b. In our example,
the correlation coefficient a would be shifted by ∆a/a = +0.015(50)% even without
electronic noise. However, the electronic noise count rate has a minor impact on the
extracted value for the neutrino-electron correlation coefficient a than the electronics
dead time. To minimize the remaining shift of the correlation coefficient a after correction
for the dead time, one has to choose the event length as short as possible and to reduce
the noise as much as possible.
As mentioned above, the secondary focus of our latest beam time at the ILL was on
the determination of a new value for a with a total relative error well below 4%. Thus,
the change of the angular correlation coefficient a by the dead time can be sufficiently
corrected, i.e., to ∆a/a < 0.1%, as long as the dead time is precisely known. Figure 5.5
illustrates the influence of a wrongly corrected dead time on the extracted value of the an-
gular correlation coefficient a. In our example, the neutrino-electron correlation coefficient
a would be shif(ted by )
∆a/a = +0.18−0.11 ± 0.05 % (5.14)
(see also Fn. 4) if the actual dead time τdead = 5.2µs would be unknown by ±0.2µs.
In Eqs. (5.9) and (5.10) it is assumed that the events are occurring randomly, i.e.,
obey Poisson statistics. But free neutron decay is a three-body decay in which the decay
products, proton, electron, and anti-neutrino, are emitted simultaneously. In our experi-
ment, only protons and electrons can be detected. If the electrostatic mirror is switched
off, decay protons are detected only if they were emitted towards the detector or if they
5.2. EXTRACTION OF A FROM THE PROTON SPECTRA 103
Figure 5.5: Relative change of the angular correlation coefficient a for a wrongly corrected dead
time. If the actual dead time τdead = 5.2µs would be unknown by ±0.2µs, the
neutrino-electron correlation coefficient a would be shifted by +0.18%−0.11% (see also Fn. 4).
Input data for the simulation (with Nachtmann’s formula Eq. (2.38)): N0 = 490 s−1,
UA = 50, 250, 400, 500, 600V, U −117 = −15 kV, Nnoise = 500 s , and a = −0.103 [10].
The error bars, in the order of 0.05%, represent the error by fitting Eq. (3.32) to the
simulated proton count rates only.
were reflected by the magnetic mirror in the DV (see Fig. 3.16a), else all decay protons
are detected. Decay electrons are detected only if they were emitted towards the proton
detector and if they can overcome the electromagnetic mirror right in front of the detec-
tor. Here, we neglect effects regarding backscattering of electrons inside the spectrometer
(discussed in Sec. 5.4.2). Due to different emission momenta and particle masses, the
time-of-flight (TOF) of the decay electrons is about one thousands of the TOF of the
decay protons. Therefore, the detection time difference between correlated coincidence
events may be estimated by the TOF of the proton. The minimum TOF of decay protons
is about 5.3µs for UA = 50V (see Dalitz plot5 5.6a and also Fn. 27 in Chap. 3) up to 8.2µs
for UA = 600V, where the proton was emitted towards the proton detector, opposite to its
correlated electron as can be seen from Fig. 5.7. If we exclude backscattering of electrons
from the bottom of the spectrometer (discussed in Sec. 5.4.2), these protons will not be
detected in coincidence with the electron from the same decay. The minimum TOF of
decay protons detected in coincidence with their correlated electron is about 7.2µs for
UA = 50V (see Fig. 5.6b) up to 10.0µs for UA = 600V, where the proton was emitted
towards the electrostatic mirror, opposite to its correlated electron. Altogether, we can
assume that the events obey Poisson statistics, provided the dead time of the electronics
is smaller than the minimum TOF of the decay protons, i.e., τdead ≤ 5.2µs.
5In high-energy physics, a Dalitz plot originally is a scatterplot of the squared invariant mass of a
subset of the particles emitted in a three-body decay, versus the squared invariant mass of another subset.
104 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.6: Dalitz plot distribution of proton time-of-flight (TOF) and polar emission angle, for
(a) protons not detected in coincidence with their correlated electron and (b) corre-
lated coincidence events. If we neglect effects regarding backscattering of electrons
(discussed in Sec. 5.4.2), a maximum of 13.1% of the decay electrons can be detected
in coincidence with their correlated proton (see Sec. 5.4.2). Since the TOF of the
electrons is about one thousands of the TOF of the protons, the detection time dif-
ference between correlated coincidence events may be estimated by the TOF of the
proton. The minimum TOF of decay protons is about 5.2µs for (a) and 7.1µs for
(b). Input data for the MC simulation (in INM approximation): U1 = U1b = 800V,
U2 = 1000V, U8 = −525V, UA = 0V, U16 = −2 kV, U17 = −15 kV, number of gen-
erated events = 107, and a = −0.105 (derived from λ = −1.2701(25) [10]). Except
for the settings of the lower dipole electrode, the input data correspond to the data
set 19_05_08/night. Here, we neglect the spatial separation of protons and electrons
achieved by both, the lower and upper, dipole electrodes.
Figure 5.7: Dalitz plot distribution of electron and proton polar emission angles. If the proton is
emitted towards the proton detector (θp = 0◦) or towards the electrostatic mirror (θp =
180◦) the electron tends to be emitted opposite. Input data for the MC simulation
(in INM approximation): Number of generated events = 109 and a = −0.105 (derived
from λ = −1.2701(25) [10]).
5.2. EXTRACTION OF A FROM THE PROTON SPECTRA 105
(a) (b)
Figure 5.8: Subtraction of the background from the pulse height spectra: (a) Typical pulse height
spectra for one detector pad from 19_05_08/night for different barrier voltages, de-
termined by adding all measurements with the same barrier voltage. With increasing
barrier voltage the count rate in the proton peak (right peak) decreases whereas the
electronic noise (left peak) is not influenced. (b) Pure proton pulse height spectra,
obtained by subtracting the background measured at UA = 780V from the measure-
ments at lower barrier voltages. The tiny peak centered around ADC channel 750 is
an artifact of the detector (for details see the text and Refs. [34, 52]). Error bars show
statistical errors only.
5.2.2 Background Correction
Figure 5.8a shows typical pulse height spectra for different barrier voltages. The left peak
below ADC channel 100 is due to electronic noise, whereas the right peak centered around
ADC channel 550 is due to protons. For systematic studies on the electronic noise [52],
the trigger settings are normally chosen in such a way that some noise is recorded. The
count rate in the proton peak decreases as the barrier voltage is ramped up and vanishes
at UA = 780V; the latter as the endpoint of the proton recoil spectrum is at about
751 eV. For UA = 780V it can be clearly seen, that the electrons and gammas constitute
an almost constant distribution over the entire visible part of the pulse height spectra.
With increasing barrier voltage less protons can pass the AP, but with higher kinetic
energy. Therefore, the maximum of the proton peak rises slightly from ADC channel 552
for UA = 50V to ADC channel 567 for UA = 600V. The tiny peak centered around ADC
channel 750 was not expected and has been carefully investigated. It is an artifact of the
detector, probably caused by a slightly higher amplification on a small spot of the detector
[34, 52].
The background is measured at UA = 780V. The pure proton count rate is obtained
by subtracting the background from measurements at lower barrier voltages. There are
two different approaches to subtract the background:
Closest: From each single measurement the count rate of the closest background mea-
surement in time is subtracted.
Averages: For each barrier voltage, the average count rate is calculated by adding all
measurements with the respective voltage. Then the average background count rate
106 CHAPTER 5. DATA ANALYSIS
is subtracted from the count rates at lower voltages. The result is shown in Fig. 5.8b.
The first method has the advantage of correcting short-term fluctuations in the count
rate. However, it is difficult not to introduce a systematic effect: If some background
measurements are used more often than others, the errors become correlated and different
weighting of the measurements will be necessary. Although the background is measured
regularly every fourth file, some measurements had to be excluded because of malfunctions
of the DAQ system. Moreover, no short-term fluctuations in the background count rate
were found in a detailed analysis of the background (see Sec. 5.3 for details). For both
these reasons, the second method, the subtraction of averages, was chosen for further
analysis.
5.2.3 Integration of the Count Rate
For each barrier voltage, the proton count rate has to be determined from the background
subtracted pulse height spectra, as shown in Fig. 5.8b. For this purpose, the count rates
are integrated over the pulse height, namely within certain limits. Normally, the lower
integration limit is set as close as possible to the electronic noise at ADC channel 80,
whereas the upper integration limit is set significantly above the proton peak at ADC
channel 1200. Obviously, the integration includes the tiny peak centered around ADC
channel 750, as the events in that peak have normal pulse shapes and as the count rate
in the tiny peak behaves exactly the same as in the main proton peak.
Figure 5.9 shows the corresponding integral proton spectrum, together with the pre-
diction from the Standard Model with the recommended value for the neutrino-electron
correlation coefficient a = −0.103 [10]. To obtain the angular correlation coefficient a,
function Eq. (3.32) is fitted to the integral proton spectrum, where wp,C,α(T ) is given
by Eq. (2.43)6. The fit parameters are the full proton decay rate N0 and of course the
angular correlation coefficient a. Finally, several different corrections have to be applied,
as discussed in the following sections.
5.3 Background
As mentioned earlier on page 38 under “First Measurements with aSPECT at the FRM II”
(see also page 56 under “Penning Traps and Penning Discharge”), in our first beam time
at the Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), situation- and time-
dependent behavior of the background was the main problem [33, 49] (see also Fig. 2.15).
To considerably improve the background conditions, the proton detector was replaced by
a silicon drift detector (SDD) at significantly reduced acceleration potential (see Sec. 3.3.1
for details), parts of the electrode system were redesigned (discussed in Sec. 3.2.1), and
the ultra high vacuum (UHV) conditions were improved (see Sec. 3.4.3).
5.3.1 Dependence on the Barrier Potential
In our first beam at the FRM II, the background count rate without neutron beam at
UA = 780V was higher than that at UA = 50V by about 3Hz [33, 49], as can be seen
from Fig. 5.10. If the actual background count rate would be unknown by this amount,
6Please note that the fit function was incorrectly described in Refs. [34, 49, 52, 197].
5.3. BACKGROUND 107
Figure 5.9: Top: Integral proton spectrum after only subtraction of the background, corresponding
to Fig. 5.8b. The green line is the prediction from the SM with the recommended value
for a = −0.103 [10], the red line shows how a deviation from that would look like.
Bottom: The black diamonds are the fit residuals. For better distinction, the green
triangles are shifted by +10V. They represent the difference between the experimental
data and a simulation with a = −0.103. Further input data for the simulation: N0 =
490 s−1 and BA/B0 = 0.203. Error bars show statistical errors only, and do not include
the uncertainty of the recommended value for a.
Figure 5.10: Background count rate without neutron beam versus barrier voltage UA. In this case,
the background count rate does not include the contribution of decay electrons. The
red squares stem from the worst data set 2604N of our first beam time at the FRM
II [33, 49], whereas the green circles stem from 20_05_08/lunch (shutter status 5) of
our latest beam time at the ILL. The comparison between the red and the green data
points shows that, for our latest beam time at the ILL, the barrier voltage dependent
background could be effectively suppressed by about an order of magnitude. However,
this still corresponds to a shift in a at the level of −1% (see also [34]).
108 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.11: Relative change of the angular correlation coefficient a for (a) different uncertainties
and (b) different slopes of the background count rate. The comparison between our
first beam time at the FRM II (red squares) and our latest beam time at the ILL
(green circles) shows that the dependence of a both on the uncertainty and the slope
of the background count rate decreases already with increasing full proton decay rate
N0, by about a factor of 2.3. We note that, for a further beam time, we have to know
the background count rate with an accuracy of 10mHz, in order to keep systematic
uncertainties in a below ∆a/a = 0.1%. In the case of a linear relationship between the
barrier voltage UA and the background count rate instead, we have to know the slope
with an accuracy of 40µHzV−1. Input data for the simulation: BA/B0 = 0.2028,
UA = 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600V for our beam time at the
FRM II [33] respectively BA/B0 = 0.203, UA = 50, 250, 400, 500, 600V for our beam
time at the ILL, and the recommended value for a = −0.103 [10]. The error bars
represent the error by fitting Eq. (3.32) to the simulated proton count rates only.
the neutrino-electron correlation coefficient a would be shifted by ∆a/a ≈ −80%, as can
be seen from Fig. 5.11a (see also Fig. 2.15b). Assuming a linear relationship between
the barrier voltage UA and the background count rate instead, a would be shifted by
∆a/a ≈ −25%, cf. Fig. 5.11b. This fact prevented us from presenting a new value for the
angular correlation coefficient a from this beam time.
In addition to the improvements mentioned above, for our latest beam time at the
ILL, an additional neutron shutter was installed inside the casemate (see Sec. 4.1.2 for
details). While this shutter has no direct influence on the background conditions, it allows
for detailed studies of the background for closed neutron shutter, i.e., without neutron
beam (e.g., in shutter status 1; defined in Sec. 4.2.4) and also shortly after measurements
with neutron beam (in shutter status 5, see also Fig. 5.13). As can also be seen from
Fig. 5.10, the AP voltage dependent background without neutron beam could be effectively
suppressed by about an order of magnitude. But if we would not correct for the measured
AP voltage dependent background, the neutrino-electron correlation coefficient a would
still be shifted by ∆a/a ≈ −2%; assuming an uncertainty of the background count rate
of 0.2 s−1 (cf. Fig. 5.11a).
We would like to emphasize, that, for a further beam time, we have to know the back-
ground count rate with an accuracy of 10mHz, in order to keep systematic uncertainties
in a below ∆a/a = 0.1% (cf. Fig. 5.11a). It is also for this reason that we have carefully
examined the background for shutter statuses 1 and 5, i.e., for closed neutron shutter
5.3. BACKGROUND 109
(a) (b)
Figure 5.12: (a) Typical pulse height spectra for one detector pad after measurements with neutron
beam (in shutter status 5). The one to two tiny peaks are visible only in logarithmic
representation after adding all measurements with the same barrier voltage. The
light blue and pink lines are Gaussian fits to these peaks for UA = 50V and 780V,
respectively. (b) Simulated pulse height spectra for different residual gas ions, with
an impact angle and kinetic energy of 0° and 15 keV, respectively. Figure taken from
Ref. [52]. The comparison between (b) and (a) shows that the simulation results are
inconclusive [52]. For details see the text. Error bars show statistical errors only.
before and after a measurement with neutron beam, respectively. This allows us to dis-
tinguish between background caused by the neutron beam, the neutron decay products,
or by other sources like field emission near the proton detector (for details see page 56
under “Penning Traps and Penning Discharge”). In contrast to our first beam time at the
FRM II (cf. the “proton-like peak” in Fig. 2.15a), only a small background contribution
was found. As can be seen from Fig. 5.12a, this contribution is visible only in logarithmic
representation after adding all measurements with the same barrier voltage. Depending
on the barrier voltage, one to two tiny peaks appear in the pulse height spectra for shutter
status 5. A detailed analysis of both peaks revealed [34]:
Peak 1 (centered around ADC channel 400): The first peak is visible both before and af-
ter measurements with neutron beam (shutter statuses 1 and 5), for both −10 kV
and −15 kV acceleration potential, and for all barrier voltages. Its maximum and in-
tensity both rise slightly with increasing barrier voltage, but its count rate fluctuates
randomly.
We believe that this background contribution is strongly correlated to the electron
trap at the AP, i.e., the trap between the lower dipole electrode e8 and the high
voltage electrodes e16 and e17 (see also Fig. 3.10).
Peak 2 (centered around ADC channel 550): The second peak is at the same position
as the proton peak (cf., e.g., Fig. 5.8b). In contrast to peak 1, it appears only
after measurements with neutron beam (shutter status 5), for −15 kV acceleration
potential, for UA = 0V and 50V, its intensity decreases from UA = 0V to 50V, and
its count rate decreases exponentially with a time constant of 2 to 3 s. As discussed
in the following section, this peak also appears randomly when the neutron shutter
is open (shutter status 3); then its count rate fluctuates non-statistically.
110 CHAPTER 5. DATA ANALYSIS
We believe that this background contribution is strongly correlated to trapped elec-
trons, which in turn may ionize residual gas molecules and/or interact with trapped
decay protons.
In Ref. [34] it was derived that background peak 1 yields a shift in the neutrino-electron
correlation coefficient a of
∆a/a = 1.14(30)% (5.15)
while peak 2 corresponds to a shift in a of
∆a/a = −2.07(31)% (5.16)
To further identify the types of ions involved in this background contribution, pulse
height spectra for possible contributors have been simulated (see also Fn. 29 in Chap. 3).
The comparison between Figs. 5.12b and 5.12a shows that the simulation results are
inconclusive [52]. Peak 2 might be caused by protons or helium ions, whereas heavier
residual gas ions, like oxygen and nitrogen, are expected below peak 1. An explanation
for peak 1 could possibly be protons with a lower impact energy of about 12 keV. However,
such protons must be produced at a potential of about -3 kV, i.e., near the high voltage
electrodes e16 or e17.
5.3.2 Dependence on the Lower Dipole Potential
As mentioned earlier on page 59 under “The Dipole Electrodes” (see also Sec. 3.4.3 under
“Elastic Scattering”), the lower dipole electrode e8 serves to remove all decay protons
and positive ions with too low energy to overcome the potential barrier and that would
otherwise be trapped between the electrostatic mirror and the AP (see also Fig. 3.10).
To investigate the efficiency of the lower dipole electrode, the temporal evolution of the
background count rate was examined for different settings of the electrode. For this
purpose, the potential barrier was set to UA = 780V and the drift potential was reduced
in several steps from U8R|U8L = −50V|−1000V to 0V|−2.5V.
Figure 5.13 shows a comparison of the temporal evolution of the background count
rate for two different settings of the lower dipole electrode. Before the neutron shutter
is opened, we only observe some counts of the environmental background (for details see
Sec. 3.4.4 under “Uncorrelated background”), independent of the drift potential. But
as soon as the shutter is opened, decay protons get trapped for the lowest potential
difference of 2.5V. Then the count rate starts to fluctuate non-statistically and does not
stop to fluctuate even after the shutter is closed again. For a drift potential of U8R|U8L =
0V|−200V instead, the count rate is stable as long as the neutron shutter is open and
drops back to the environmental background count rate immediately after the shutter is
closed again.
We note that for the lowest drift potential of U8R|U8L = 0V|−2.5V, an additional peak
centered around ADC channel 230 is visible in the pulse height spectra for shutter status
5 [34]. The additional peak might be caused by heavier residual gas ions, like oxygen
or nitrogen (cf. Fig. 5.12b), but it can be effectively suppressed by the typical potential
difference of 200V.
To further investigate possible background contributions from electrons, the lower
dipole electrode was used as a second analyzing plane. More precisely, a positive potential
5.3. BACKGROUND 111
Figure 5.13: The temporal evolution of the background count rate measured at UA = 780V for
two different settings of the lower dipole electrode e8, from 19_05_08/lExB. For
elucidation, the dashed blue lines show when the neutron shutter was opened and
later closed again. The comparison between the red and the green data points shows
the efficiency of the lower dipole electrode in the emptying of the possible positive
ion trap between electrostatic mirror and analyzing plane (cf. Fig. 3.10). Error bars
show statistical errors only.
(a) Shutter status 3 (b) Shutter status 5
Figure 5.14: Pulse height spectra for one detector pad from 03_05_08/lExB_p1kV for a second
analyzing plane. For this investigation a positive potential of +1000V was applied to
both sides of the lower dipole electrode e8. Depending on the barrier voltage UA, one
to two peaks appear in the pulse height spectra for (a) measurements with neutron
beam, while only one tiny peak appears (b) after measurements with neutron beam.
We note that in the case of (a) the background count rate includes the contribution of
decay electrons. In comparison with Fig. 5.12, these spectra stem from measurements
at an acceleration potential of −10 kV and therefore both background peaks are
shifted to lower ADC channels. But in contrast to Fig. 5.12a and Sec. 5.3.1, for
measurements with neutron beam, the second peak is visible for all barrier voltages.
Error bars show statistical errors only.
112 CHAPTER 5. DATA ANALYSIS
of +1000V was applied to sides R and L of the dipole electrode, in order to prevent all
decay protons from reaching the AP. Figure 5.14 shows the corresponding pulse height
spectra for different barrier voltages UA. Again, one to two peaks appear in the pulse
height spectra for measurements with neutron beam, while only one tiny peak appears
after measurements with neutron beam. In contrast to Sec. 5.3.1, for measurements with
neutron beam, the second peak is visible for all barrier voltages. In this case, the derived
shift in the angular correlation coefficient a Eq. (5.16) would be wrong. However, we must
also note that these pulse height spectra were recorded at a time when the count rates
were fluctuating non-statistically7 and with manually switching of the electrostatic mirror
only. Hence, we should not jump to conclusions until there is evidence that the second
background peak is present for all barrier voltages. And since the second peak is at the
same position as the proton peak, we strongly recommend to repeat this measurement in
a further beam time, but with automated switching of the electrostatic mirror. Where
possible, the second analyzing plane should be applied to different electrodes both below
and above the AP electrode e14.
5.3.3 Trapping of Decay Protons
As discussed in greater detail later in Sec. 6.4.3, work function inhomogeneities in the
DV could lead to trapping of decay protons between the DV and the electrostatic mirror.
Approaches to eliminate and/or to handle this effect are presented in Sec. 6.4.4. Due to a
lack of time during our latest beam time at the ILL, we could only study different “Inverse
electric field gradient[s]”. For the investigation of electric field gradients, a potential of a
few volts is applied to the bottom and top cylinders e3 and e6 of the DV electrode, with
opposite sign. Only in the case of a positive gradient, decay protons may be trapped be-
tween the DV and the electrostatic mirror, whereas in the case of a negative gradient, the
protons remain unaffected. In the event of work function inhomogeneities, described by a
positive electric field gradient, a superimposed negative gradient should reduce the trap-
ping conditions. To be specific, with increasing negative electric field gradient the proton
count rate should increase, until the work function inhomogeneities are compensated.
Figure 5.15 shows the short-term evolution of the proton count rates for two differ-
ent settings of the DV electrodes e3 and e6. Absolutely unexpected the proton count
rates, both for UA = 0V and 50V, decrease with increasing negative electric field gra-
dient. Obviously, these measurements were performed at a time when the count rates
were fluctuating non-statistically (cf. Fn. 7). However, the count rate losses of several
Hz (cf. Table 5.3) are so large that another effect seem to be in play. In the event that
the electrical connections of the electrodes e3 and e6 have been changed by mistake, the
count rate losses might at least partly be explained by an additional electric mirror above
the DV, as can be seen from Table 5.3. It is very likely that we investigated different
“Definite electric mirror[s] above the DV” (cf. Sec. 6.4.4) instead of “Inverse electric field
gradient[s]”. Hence, we recommend to repeat this measurement in a further beam time.
7At the beginning of our latest beam time at the ILL, the count rates were fluctuating non-statistically.
We identified and fixed problems with a power supply for side R of the lower dipole electrode e8, several
ground loops, and the electrical connections of the electrode system.
5.3. BACKGROUND 113
(a) U3 = −U6 = 1V (b) U3 = −U6 = 2V
Figure 5.15: Influence of an electric field gradient in the decay volume (DV) on the proton
count rates from 23_04_08/DV_gradient_evs100 for different barrier voltages UA =
0, 50, 780V. For this investigation a potential of a few volts (a) U3 = −U6 = 1V re-
spectively (b) U3 = −U6 = 2V was applied to the bottom and top cylinders e3 and
e6 of the DV electrode. In comparison with the top of Fig. 5.9, these count rates
stem from a measurement at an acceleration potential of −10 kV and are therefore
lower by about 15 to 20Hz. Unfortunately, the measurement results are inconclusive
(see also Fn. 7); see the text and Table 5.3 for details. Error bars show statistical
errors only.
Table 5.3: Influence of an electric field gradient ∂U0/∂z in the decay volume (DV) on the proton
count rate for UA = 50V, corresponding to Fig. 5.15 and Eq. (6.19). For this purpose
a potential of a few volts, U3 = −U6, was applied to the DV electrodes e3 and e6. In
the event of work function inhomogeneities in the DV, the count rates should increase
with the superimposed external electric field, by a few Hz (cf. the fourth column).
This is not the case. Hence, we assume that the electrical connections of the electrodes
e3 and e6 have been changed by mistake. The comparison between the third and the
last column shows that this might at least partly explain the count rate losses by an
additional electric mirror above the DV, corresponding to Eq. (3.30). Input data for
the calculation (with Nachtmann’s formula Eq. (2.38)): B0 = 2.177T, BA/B0 = 0.203,
and the recommended value for a = −0.103 [10]. The following abbreviations appear:
avg (average value) and CPS (count rate or counts per second).
U3 = avg avg CPS CPS due to electric CPS due to electric
−U6 ∂U0/∂z measured field gradient mirror above DV
[V] [mVcm−1] [s−1] [s−1] [s−1]
0 0 448.0± 1.0 448.0 448.0
-1 25 440.3± 1.4 446.8 441.6
-2 50 431.5± 1.0 444.9 427.6
5.3.4 Trapping of Decay Electrons
Due to the slight magnetic field gradient (∂B0/∂z) /B −4 −10 = −1 × 10 cm in the DV,
a small fraction of decay protons and electrons, emitted in the negative z-direction with
114 CHAPTER 5. DATA ANALYSIS
(initial) polar angle 90°< θ0 <92°, is reflected by the magnetic mirror just below the
DV (see Fig. 4.5b). While all protons can overcome the magnetic mirror right in front
of the proton detector (see Fig. 4.5a), only about 13.1% of the electrons will reach the
detector (for details see Sec. 5.4.2). In particular, decay electrons with (initial) polar
angles 88°< θ0 <92° will be reflected both on the magnetic mirror just below the DV and
right in front of the proton detector.
Assuming a unifo∫rm neutron beam density, a fraction of about8
1
1 − ∫ dP0wtr(T0; P0)
P dP00 ∫ P0 √
(3.29) ∫ 1 − B0 − z · 9.25× 10−50 ·B= 0∫ dz0 √1z dz0 z B0 + 6 · 9.25× 10−5 ·B0 00∫ √4 ∫1 (z0 + 6) · 9.25× 10−5 1 92.5× 10−3 10 √= 4 dz0 − = √ dz0 z5 0
− dz0 −4
1 + 6 · 9.[25× 10] 8 1 + 6 · 9.25× 10−5 2√ 4
92.5 10−3 2 10
= √ 3/2z
8 1 + 6 · 9 0√ .25× 10
−5 3 2
92.5 10−3 ( )
= √ 103/2 − 23/2 (5.17)
12 1 + 6 · 9.25× 10−5
≈ 2.3%
of all decay electrons will therefore be trapped between the DV and the upper dipole
electrode e16. Here, P0 = (x0, y0, z0) denotes the decay point and B0 is the average
magnetic field in the DV (see also Sec. 4.1.4). For a full proton decay rate of N0 = 490 s−1
this corresponds to about 12Hz. These electrons perform lots of axial oscillations between
the DV and the upper dipole electrode before they have lost their total energy by means
of inelastic scattering on residual gas molecules and/or synchrotron radiation. The latter
process can take up to 1 s [224], assuming a constant magnetic field of 1T. For a residual
gas pressure of 1×10−9 mbar, one such trapped decay electron would produce background
of about 10Hz [224]. The exact background count rate depends on its storage time.
In Ref. [47] (see also Sec. 3.4.4 under “Uncorrelated background events”) it is assumed
that this background contribution does not depend on the barrier voltage and can therefore
be measured at UA = 780V. Indeed these electrons spend most of their time at their both
turning points, i.e., around the DV and below the upper dipole electrode. Hence, it is less
than likely that these electrons interact with residual gas molecules in the AP, in which
case this background contribution would depend on the barrier voltage.
To further investigate this systematic effect, we recommend measurements with dif-
ferent magnetic field gradients, in a further beam time. In particular, we propose a mea-
surement with a positive magnetic field gradient, i.e., with a magnetic mirror above the
DV, so that no decay electrons can be trapped above the DV. We note that this investi-
gation is not independent of possible work function inhomogeneities in the DV (discussed
in Sec. 6.4.4 under “Definite magnetic mirror above the DV”).
A detailed analysis of the (measured) background conditions can be found in Ref. [34]
(see also [52]).
8In our measurement, we expect a fraction of only about 2.10± 0.01(stat)+0.04−0.01(sys)%, cf. Eq. (5.52).
5.4. UNEXPECTED SYSTEMATIC EFFECTS 115
(a) Full view (b) Zoom
Figure 5.16: The uncorrected value aexp of the angular correlation coefficient a for (a) different
lower integration limits, for one detector pad from 16_05_08/night. The (b) zoom
to values aexp = −0.1 ± 0.03 shows that, for lower integration limits higher than
400ADC channels, the steep rise of aexp disappears after correction by simulation
of the proton recoil spectrum, whereas the flat slope, for lower integration limits
smaller than 340ADC channels, remains unaffected. The error bars show statistical
errors only and are strongly correlated, as each data point stems from the same
measurement run. Figures taken from Ref. [52].
5.4 Unexpected Systematic Effects
As mentioned earlier in Chap. 4, the primary focus of our latest beam time at the ILL
was on the identification and investigation of possible systematic effects as discussed in
Sec. 3.4 (see also Sec. 4.2.5). However, among other unexpected systematic effects, we
identified and fixed a problem in the detector electronics. In this (see also Sec. 6.4.3) and
the following Sec. 5.5, we discuss the unexpected and “expected” systematics, respectively.
We note that the values given in this and the following Sec. 5.5 for the neutrino-electron
correlation coefficient a are for comparison purposes only. We would like to stress that
these values do not represent a final value for a, as they are corrected only for the dead
time of the electronics Eq. (5.10) and the blind analysis Eqs. (5.1) and (5.2).
5.4.1 Dependence of a on the Lower Integration Limit
During the data analysis, both the upper and the lower integration limit were investigated
and a rather strong dependence of the angular correlation coefficient a on the lower inte-
gration limit was found. From a lower integration limit of 80 to 340ADC channels, the
uncorrected value for a decreases by
∆aexp/aexp = (−9.6± 4.7)% (5.18)
from aexp = −0.094(3) to aexp = −0.103(3), as can be seen from Fig. 5.16. For lower
integration limits higher than 400ADC channels, the value for aexp increases steeply to-
wards positive values. In the latter area, the strong increase of aexp can be explained by
the electric retardation method of aSPECT and can therefore be corrected by simulations
(for details see [52] and also Fn. 29 in Chap. 3): As already mentioned in Sec. 5.2.2, with
116 CHAPTER 5. DATA ANALYSIS
increasing barrier voltage UA less protons can overcome the AP, but with higher kinetic
energy. Thus, for a fixed lower integration limit for all barrier voltages, one cuts away
more counts for the lower than for the higher voltages, as can be seen from Fig. 5.8.
Figure 5.16 also shows that, for lower integration limits higher than 340ADC channels,
the steep rise of aexp disappears after correction by simulation of the proton recoil spec-
trum. However, for lower integration limits smaller than 340ADC channels, the flat slope
remains unaffected. Obviously, the remaining dependence must be caused by a completely
different systematic effect, as will be discussed in the following sections. We note that, for
a lower integration limit of 80ADC channels, the correction is only
∆a/a = 0.13(9)% (5.19)
with a relative error of 70% dominated by simulation statistics [52].
For a fixed lower integration limit, no such dependence of the angular correlation
coefficient a on the upper integration limit was found, even if the upper integration limits
falls below the tiny peak centered around ADC channel 750 (see Fig. 5.8).
Details on the selection of upper and lower integration limit can be found in Ref. [34]
(see also [52]).
Correlated Events
To further investigate the dependence of the neutrino-electron correlation coefficient a on
the lower integration limit, we analyzed our measurement data for possible anomalies. This
revealed a time-dependent loss mechanism of decay protons shortly after high-energy elec-
trons [52]. Figure 5.17a shows pulse height spectra of events after high-energy electrons,
sorted according to their time difference to the high-energy electron. For pulse-heights
below ADC channel 200,
• there are almost no events less than 8µs after a high-energy electron,
• while obviously “additional” events between 8 and 15µs after a high-energy electron.
Thus, we probably lose decay protons with low pulse height and short time difference
to the electron, as described in the following section.
As discussed earlier in Sec. 5.2.1, the detection time difference between correlated
coincidence events may be estimated by the TOF of the proton. The minimum TOF of
decay protons is about 5.3µs for UA = 50V (see also Fn. 27 in Chap. 3) up to 8.2µs for
UA = 600V, where the correlated electron was emitted towards the electrostatic mirror.
If we neglect effects regarding backscattering of electrons (discussed in Sec. 5.4.2), these
protons will not be detected in coincidence with the electron from the same decay. The
minimum TOF of decay protons detected in coincidence with the correlated electron is
about 7.2µs for UA = 50V up to 10.0µs for UA = 600V. Thus, for UA ≥ 500V, we
cannot lose decay protons, as can also be seen from Fig. 5.17b. Hence, the loss rate
strongly depends on the barrier voltage UA, what, in turn, has a major impact on the
angular correlation coefficient a.
However, the question remains whether both above mentioned effects can explain the
observed shift Eq. (5.18) in the uncorrected value for a. As can be seen from Fig. 5.18, the
energy distribution of only correlated coincidence events is shifted to lower proton kinetic
5.4. UNEXPECTED SYSTEMATIC EFFECTS 117
(a) (b)
Figure 5.17: (a) Background subtracted pulse height spectra of events after high-energy electrons
from 20_05_08/night for UA = 50V. According to their time difference to the elec-
tron, the events were sorted into three different categories. Apparently, we lose decay
protons with low pulse height (ADC channels < 200) and short time difference to
the electron (black points). We note that the spectra were normalized to the count
rate between ADC channels 500 and 600. (b) Typical (detection) time difference
spectra from 18_05_08/night for different barrier voltages UA. With increasing
barrier voltage the count rate decreases whereas the minimal TOF increases. The
colorful highlighted areas correspond to the three different categories from (a). For
UA ≥ 500V, we cannot lose decay protons, as their minimal TOF is greater than
8µs. Thus, the loss rate strongly depends on the barrier voltage. Error bars show
statistical errors only.
(a) (b)
Figure 5.18: Dalitz plot distribution of electron and proton kinetic energy, for (a) protons not de-
tected in coincidence with their correlated electron and (b) correlated coincidence
events. Please note the different contour scales. If we neglect effects regarding
backscattering of electrons (discussed in Sec. 5.4.2), a maximum of 13.1% of the decay
electrons can be detected in coincidence with their correlated proton (see Sec. 5.4.2).
Input data for the MC simulation (in INM approximation): Number of generated
events = 109 and a = −0.105 (derived from λ = −1.2701(25) [10]).
118 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.19: (a) A proton event shortly after an high-energy electron. The first (electron) event
saturates the detector electronics and is therefore cut off by the maximum ampli-
fication of the electronics. After the first (electron) event the baseline drops down
to around 1100 ADC channels. The next (proton) event therefore has a lower and
changing baseline. We note that the red data points show the ADC values of two
consecutive events, where the values between the two events are not recorded. (b)
Typical baseline value distribution for one measurement file. Most events have a base-
line of (1490± 20)ADC channels, whereas events shortly after high-energy electrons
have a lower baseline of down to 1000 ADC channels. Error bars show statistical
errors only.
energies. Analysis of only correlated coincidence events (Fig. 5.18b) would therefore lead
to a = +0.546, whereas analysis of only protons not detected in coincidence with their
correlated electron (Fig. 5.18a) would lead to a = −0.237. Hence, the observed shift
Eq. (5.18) is realistic.
Baseline Shifts
Our analysis revealed that the trigger efficiency is reduced after high-energy electrons [52],
what at least partly explains the unexpected proton losses less than 8µs after high-energy
electrons.
As mentioned earlier in Sec. 5.1.3, the pulse of an high-energy electron is longer than
one event window (5µs). For a proton shortly after an high-energy electron, it is therefore
probable that the proton will sit on the decaying edge of the electron, as shown in the top
of Fig. 5.2b. In most cases, the baseline of such a proton is changed by the exponentially
decaying edge of the electron, as can be seen in Fig. 5.19a (see also Fig. 5.2b). Figure 5.19b
shows that most events have a baseline of (1490±20)ADC channels, whereas events shortly
after high-energy electrons have a lower baseline of down to 1000 ADC channels.
The pulse heights of the measured events can be corrected with the adapted fit (de-
scribed in Sec. 5.1.3). However, the shifts of the baseline already influence the data
acquisition by means of the trigger algorithm (described on page 69 under “Digital Elec-
tronics and the Trigger Algorithm”): On the decaying edge of an high-energy electron, the
value of the (baseline) window w1 is shifted to higher values compared to the true baseline
value at the position of trigger window w2. If the shift of the (baseline) window w1 is
too high, events with a rather low pulse height will be lost. In other words, the trigger
efficiency is reduced after high-energy electrons.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 119
(a) (b)
Figure 5.20: (a) The reduced trigger ranges after high-energy electrons for a baseline lowered to
1000 ADC channels and different endpoints of the saturation. The scatter plots show
that the trigger efficiency decreases with later endpoint of the saturation. For elucida-
tion, the horizontal and the vertical blue line indicate the dead time of the electronics
(5.2µs), i.e., the minimal time difference to the electron, and the usually used lower
integration limit (80ADC channels), respectively. Figure taken from Ref. [52]. (b)
Typical relationship between the pulse height of protons after high-energy electrons
and the lowered baseline value, for one measurement file with UA = 50V. The broad
(turquoise) band shows the distribution of the proton events with a lowered baseline
(see also Fig. 5.19b), whereas the thin diagonal (blue) branch reveals the “additional”
events between 8 and 15µs after high-energy electrons (see also Fig. 5.17a).
To reproduce the proton losses less than 8µs after high-energy electrons, the reduced
trigger efficiency was investigated by means of MC simulations [52]. For this purpose
electron-proton coincidence events were generated and subsequently analyzed with the
trigger algorithm described on page 69 under “Digital Electronics and the Trigger Algo-
rithm”. To determine the reduced trigger range, the pulse height of the proton, its time
difference to the electron, the endpoint of the saturation, and the lowered baseline value
were varied. For example, Fig. 5.20a shows the reduced trigger ranges for a baseline low-
ered to 1000 ADC channels and different endpoints of the saturation. As expected, the
trigger efficiency decreases with later endpoint of the saturation.
In Ref. [34] it was derived that the reduced trigger efficiency after high-energy electrons
corresponds to a shift in the neutrino-electron correlation coefficient a of
∆a/a ≈ 0.2% (5.20)
This is far too little to explain the observed shift Eq. (5.18) in the uncorrected value
for a. Therefore the strong dependence of the angular correlation coefficient a on the
lower integration limit must be caused by a different systematic effect, as described in the
following section.
We mention that further conclusions were drawn from scatter plots, showing the rela-
tionship between the pulse height of protons after high-energy electrons and the lowered
baseline value. Figure 5.20b shows such a scatter plot. As expected, the broad (turquoise)
band shows the distribution of the proton events with a lowered baseline. However, the
figure also reveals a thin diagonal (blue) branch, from normal pulse heights at around ADC
120 CHAPTER 5. DATA ANALYSIS
channel 1050 down to almost zero pulse height at around ADC channel 1350. This branch
corresponds to the “additional” events between 8 and 15µs after high-energy electrons,
already observed in Fig. 5.17a (see also the previous section).
We note that the trigger efficiency issue can be easily solved by an optimized pulse
shaping, in a further beam time.
Saturation of the Preamplifier
Figure 5.20b revealed that the “additional” events between 8 and 15µs after high-energy
electrons are not distributed randomly but lie on a thin diagonal branch. In particular,
a slightly lowered baseline already leads to a strongly reduced pulse height. Further
investigations showed that the “additional” events appear only after high-energy electrons
which saturate the detector electronics for at least about 3µs [52]. We are convinced that
this effect is due to a saturation of the preamplifier after very high-energy events.
The principle of this saturation effect is shown in Fig. 5.21. As explained earlier
on page 68 under “Amplification Boards”, the raw signal shape consists of a steep rise
(τ 9rise ≈ 25 ns) followed by a long exponential decay (τdecay ≈ 150µs, see also Fn. 9). In
the case of a double event, a second event shortly after the first one will therefore sit on
the decaying edge of the first event. Hence, if an high-energy electron deposits so much
energy in the detector that the preamplifier comes close to its maximum amplification, a
proton shortly after the electron will drive the preamplifier into saturation. Consequently,
the proton peak is cut off. Then, the shaper differentiates and integrates the amplified
signal. The shaper is mostly sensitive to the rising edge of the signal and shortens the
pulse. And since the height of the pulse after the shaper is proportional to the change in
pulse height before the shaper, the proton peak will ultimately have a lower pulse height
after the shaper. We note that the cut-off (of high-energy events) seen in the pulse shapes
(see Fig. 5.1b) is due to a saturation of the shaper (see Fig. 6.27 in Ref. [52]).
To confirm our suspicion that the saturation of the preamplifier causes the reduced
pulse height of the “additional” events and, at the same time, to reproduce this saturation
effect, the detector electronics was tested in several steps [52]. For this purpose, the proton
detector has been replaced by a waveform generator. A high, electron-like pulse followed
by a low, proton-like pulse were generated with a waveform generator, fed through the
entire electronics chain, and subsequently analyzed with the trigger algorithm described
on page 69 under “Digital Electronics and the Trigger Algorithm”. To reproduce the
saturation effect, the amplitude of the electron-like pulse and its time difference to the
proton-like pulse were varied.
For example, Fig. 5.22 shows pulse height spectra for two different amplitudes of the
electron-like pulse and several different time differences between the electron-like and the
proton-like pulse. In contrast to, e.g., Fig. 5.8a the “left” peak below ADC channel 100, due
to electronic noise, is missing. Since the detector was disconnected from the electronics, the
noise level was much lower than usual and hence no noise was triggered this time. Spectra
for a time difference lower than the lowest one shown do not contain any proton-like
counts, due to the reduced trigger efficiency after high-energy electrons (see the previous
section for details). As expected, the saturation of the preamplifier shifts the proton-like
peaks to lower pulse heights. With decreasing time difference to the electron-like pulse
the maximum of the proton-like peaks drops to lower pulse heights. Furthermore, with
increasing amplitude of the electron-like pulse the proton-like peaks are shifted to even
9Please note that in Ref. [34] both the typical rise and the approximate decay time are misstated.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 121
(a) (b)
Figure 5.21: Illustration of the signal processing for (a) a single event and (b) two consecutive
events, where the second event drives the preamplifier into saturation. The figures
show, from top to bottom, how (top) a signal from the detector is (middle) amplified
by the preamplifier, then (bottom) shaped by the adapter board, and finally (not
shown here) digitized by the ADC (see page 68 for details). Assuming a first event
deposits so much energy in the detector that the preamplifier comes close to its
maximum amplification (blue, dashed), a second event shortly after the first one will
drive the preamplifier into saturation. Therefore, the second peak (green, dashed)
is cut off and, consequently, will have a reduced (red, solid) pulse height after the
shaper. Illustration adapted from Ref. [52].
lower pulse heights and the minimum time difference to trigger a proton-like pulse in-
creases. We mention that, in our real measurement, proton-like counts below ADC chan-
nel 80 are superimposed on the electronic noise. However, lowering the lower integration
limit would not provide additional information about the angular correlation coefficient
a, because of fluctuations in the electronic noise.
In Ref. [52] it was derived that a decay electron has to deposit a minimum energy of
T sate,deposit ≥ 400(20) keV (5.21)
in the proton detector in order to saturate the preamplifier (see also Eq. (4.8)), with a
relative error of 5% dominated by the difficulty to determine to start point of the satura-
tion effect. To determine which (with regard to their impact angle and energy) and how
122 CHAPTER 5. DATA ANALYSIS
(a) Amplitude 220mV (b) Amplitude 240mV
Figure 5.22: Pulse height spectra for two different amplitudes (in mV) of the electron-like pulse
and several different time differences between the electron-like and the proton-like
pulse. With decreasing time difference the maximum of the proton-like peaks drops to
lower pulse heights. The comparison between (a) and (b) shows that, with increasing
amplitude of the electron-like pulse, the proton-like peaks are shifted to even lower
pulse heights and the minimum time difference to trigger a proton-like pulse increases.
For better distinction, only a selection of pulse height spectra is shown. Error bars
show statistical errors only. Figures taken from Ref. [52].
many decay electrons can deposit more than 400 keV inside the detector, the backscat-
tering of electrons was investigated by means of MC simulations10 [52]. The simulations
have shown that only a fraction of the high-energy electrons is stopped inside the proton
detector. Figure 5.23a shows the fraction of electrons that deposit more than 400 keV
inside the detector, for different impact angles and energies. According to this, only about
12.6% of the decay electrons that reach the proton detector deposit more than 400 keV
inside the detector (see also Fig. 5.18b). If we neglect effects regarding backscattering of
electrons (discussed in Sec. 5.4.2), a maximum of 13.1% of the decay electrons can be
detected in coincidence with their correlated proton (see Sec. 5.4.2). Therefore, not more
than about 1.66% of all decay protons are affected by the saturation effect. However, the
relationship Eq. (5.21) might change as soon as the waveform generator is replaced by the
proton detector again.
In addition, Figure 5.23b shows that the dead time of the detector might rise up to
about 50µs after electrons that deposit more than 750 keV inside the detector. However,
during our latest beam time, we observed “additional” events only less than 20µs after
high-energy electrons [225] (see also page 116 under “Correlated Events”). But on the
other hand, it is unlikely that such high-energy electrons are stopped inside the proton
detector [52, 225].
We note that the preamplifier saturation issue can be easily solved by reducing the
amplification of the preamplifier down to about 40%, in a further beam time.
Details on the problem in the detector electronics, simulations of electron-proton co-
incidence events as well as electron energy deposition inside the proton detector (see also
10The backscattering of electrons was simulated with CASINO [226]. The CASINO v2.42 program is
a Monte Carlo simulation of electron trajectory in solid specially designed for low beam interaction in a
bulk and thin foil [227, 228].
5.4. UNEXPECTED SYSTEMATIC EFFECTS 123
(a) (b)
Figure 5.23: (a) The fraction of electrons that deposit more than 400 keV inside the proton detec-
tor, for different impact angles and energies. With increasing impact kinetic energy
this fraction decreases. Altogether, only about 12.6% of the decay electrons that
reach the detector deposit more than 400 keV inside the proton detector and hence
not more than about 1.66% of all decay protons are affected by the saturation effect.
(b) Change of the dead time due to the saturation of the preamplifier. With increas-
ing impact kinetic energy the minimum time difference to trigger a proton pulse and
hence the dead time increase (see also Fig. 5.22). The colored lines are different fits
to the measured data points, except for the blue one. See the following section for
details. For elucidation, the horizontal blue line indicates the actual dead time of the
electronics (5.2µs). Data from Ref. [225].
Fn. 10), and tests of the detector electronics can be found in Ref. [52] (see also [34]).
Correction for the Saturation Effect
At first sight, it might be thought that it is easy to correct for the saturation effect,
by simply introducing an artificially increased dead time, τart (see later in Fig. 5.25a).
With regard to counting statistics, we cannot select τart such that we cut away all protons
detected in coincidence with the electron from the same decay. Rather, we must determine
a minimal artificial dead time, so that the uncertainty in the respective shift in a is
small compared to the statistical accuracy. However, the amount of protons detected in
coincidence with their correlated electron strongly depends on the spatial separation of
electron and proton. The spatial separation, in turn, depends on the settings of all three,
the lower and the upper dipole electrodes as well as the height of the main magnetic
field (cf. Sec. 5.5.6). That is why, if we introduce an artificially increased dead time, we
have to correct for the proton count rate reduction by calculating the proton and electron
trajectories (by means of MC simulations). However, in addition to the saturation effect,
our data analysis revealed that a fraction of the decay electrons is backscattered from the
bottom of the aSPECT spectrometer (for details see the following section). Unfortunately,
a fraction of these electrons will be detected in coincidence with the proton from the same
decay. Therefore, if we introduce an artificially increased dead time, we also have to correct
for their correlated protons. Consequently, we can instead also reproduce the observed
shift in the neutrino-electron correlation coefficient a with MC simulations.
Figure 5.24a shows the flow chart of the MC simulation to reproduce both the satura-
124 CHAPTER 5. DATA ANALYSIS
(a) Complete description (b) Simplified model
Figure 5.24: Flow chart of the MC simulation to reproduce the saturation effect. (a) Complete description of both the saturation of the preamplifier
and the electron backscattering inside the aSPECT spectrometer. (b) Simplified model to describe both just mentioned effects. We
decided to go for model (b), as too many input parameters for the saturation effect could only be determined with large uncertainties
[52]. The red highlighted rhombuses and parallelograms indicate decisions and output in connection with the saturation effect.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 125
tion effect and the backscattering of decay electrons inside the spectrometer. With regard
to the saturation effect, there are far too many input parameters for the MC simulation
which could only be determined with large uncertainties. In particular, there is no correc-
tion matrix that describes the relationship between electron impact energy and angle and
the energy deposited by the electron in the proton detector (see the previous section and
also [52]). Hence, we decided to go for a simplified MC simulation instead, as shown in
Fig. 5.24b. In addition, as in the case of our MC Simulations of the edge effect (see also
page 161 under “Correction for the Edge Effect”), we decided to derive the impact of the
saturation effect on the angular correlation coefficient a from combined MC simulations,
where
• MC simulations in (2D) axially symmetric fields serve to determine the dependence
of a on the single parameters of the saturation of the preamplifier and
• MC simulations in (3D) non-axially symmetric electric fields serve to determine the
dependence of a on the saturation effect for different settings of the spectrometer,
i.e., the spatial separation of decay protons and electrons (see later in Table 5.6).
Here, the data from Fig. 5.23 are used as input for the MC simulations. We mention that
the impact of the backscattering of decay electrons from the bottom of the spectrometer
on a (in combination with the saturation effect) was determined separately (discussed in
greater detail in Sec. 5.4.2).
For our standard settings (cf. Table 3.1), but in 2D axially symmetric electric fields,
i.e., U8 = −525V and U16 = −2 kV, the saturation of the preamplifier yields a shift in the
neutrino-electr(on correlatio)n coefficient a of
∆a/a = −27.28+2.14−0.00 %, (5.22)
where the error reflects the uncertainty in fixing the relationship between the dead time of
the proton detector after an high-energy electron and the energy deposited by the electron
in the detector, cf. Fig. 5.23b.
As mentioned in the previous section, in contrast to Fig. 5.23b, we observed “addi-
tional” events only less than 20µs after high-energy electrons. From our 2D MC simula-
tions for different fixed dead times of the proton detector after an high-energy electron,
we expect an error(in the shift)in a of
∆ (∆a/a) = +3.31+0.34−1.21 %, (5.23)
in agreement with the error identified above in Eq. (5.22).
In addition, as also stated in the previous section, the minimum energy to saturate the
preamplifier Eq. (5.21) could only be derived with a relative error of 5%. From our 2D
MC simulations for different fixed minimum energies, we expect an additional uncertainty
in the shift in a of
∆ (∆a/a) = ±1.44%. (5.24)
For our standard settings, but in 3D non-axially symmetric electric fields, i.e., U8R =
−50V, U8L = −1000V, and U16A = U16B = −2 kV, the saturation effect yields a shift in
the angular correlation coefficient a of
∆a/a = (−17.54± 3.77) %, (5.25)
126 CHAPTER 5. DATA ANALYSIS
(a) Experimental data (b) 2D MC simulation
Figure 5.25: Relative change of the angular correlation coefficient a for an artificially increased
dead time, τart: (a) Relative change of the uncorrected value, aexp, for a from
19_05_08/night for two different lower integration limits. The error bars show sta-
tistical errors only and are strongly correlated, as each data point stems from the
same measurement run. (b) Relative change of a for different input parameters for
the MC simulation, according to the previous section and Fig. 5.23. Obviously, the
linear fit to only two (turquoise) instead of all three (green) data points in Fig. 5.23
has little influence on a, while a logarithmic (blue) instead of the linear fit has no
influence on a. See the text for details. Input data for the MC simulation (in INM
approximation): B0 = 2.177T, BA/B0 = 0.203, U1 = U1b = 800V, U2 = 1000V,
U8 = −525V, UA = 50, 250, 400, 500, 600V, U16 = −2 kV, U17 = −15 kV, number of
generated events = 107, and the recommended value for a = −0.105 (derived from
λ = −1.2701(25) [10]). Except for the settings of the lower dipole electrode (e8), the
input data correspond to the data set 19_05_08/night.
with a relative error of 21.5% dominated by simulation statistics (number of generated
events = 6× 106). Compared to Eq. (5.22), this corresponds to a reduction of the shift in
a due to the spatia(l separation) of decay protons and electrons of
∆ (∆a/a) = +9.54+3.77−4.34 %. (5.26)
Altogether, the comparison with our calculations in 2D axially symmetric fields results in
a shift in the n(eutrino-elect)ron correlation coefficient a due to the saturation effect of
∆a/a = −17.54+5.23−4.21 %. (5.27)
However, Eq. (5.27) only takes into consideration the change of the dead time of the
proton detector after high-energy electrons, but it does not include the shift of the proton
peaks to lower pulse heights as shown in Fig. 5.22. For this reason, Eq. (5.27) can only
serve to correct the experimental values, aexp, for a for a lower integration limit of 0 ADC
channels. Unfortunately, due to fluctuations in the electronic noise, we cannot lower the
lower integration limit to below ADC channel 80. With regard to the saturation effect, it
would not be very serious to, e.g., linearly extrapolate the uncorrected values aexp for a
from ADC channel 80 to 0.
On the other hand, we still can investigate the experimental values aexp for a for an
artificially increased dead time. Figure 5.25 shows the influence of an artificially increased
5.4. UNEXPECTED SYSTEMATIC EFFECTS 127
dead time on the experimental value for a in comparison with the expected values from
our 2D MC simulations. Obviously, the functional dependence of the relative shift in a
depends on the chosen lower integration limit and particularly on the input parameters
for the MC simulation. If we use the data from Fig. 5.23 as input for the MC simulations,
then the functional dependence is very similar to the measurement. We note that the
deviation from simulation to measurement between τart = 5 and 15µs is dominated by
the poor agreement between the measured and the simulated TOF spectra, cf. Fig. 5.26
(for details see the following section). In addition, the real deviation for τart = 30µs is
different from Fig. 5.25b, because of the spatial separation of decay protons and electrons
Eq. (5.26).
From the measurements (cf. Fig. 5.25a), we extract an absolute shift in a due to an
artificial dead tim{e τart = 30µs of
−0.0809(49) for a lower integration limit of 80 ADC ch.
∆aexp = (5.28)
−0.0772(49) for a lower integration limit of 300 ADC ch.
whereas an absolute shift in a of
∆a = −0.0881(14) (5.29)
from our 2D MC simulations (cf. Fig. 5.25b). The latter has to be corrected for the spatial
separation of decay protons and electrons by
∆a = +0.0100+0.0040−0.0046, (5.30)
cf. Eq. (5.26). All three together, we have to correct Eq. (5.27) for the shift of the proton
peaks to lower pu{lse(heights by( )−2.66+6.16−6.53) % for a lower integration limit of 80 ADC ch.∆a/a =
+0.86+6.16−6.53 % for a lower integration limit of 300 ADC ch.
(5.31)
After correction, we derive a shift in the neutrino-electron correlation coefficient a
due to the saturation of the preamplifier of
{ (( )−20.20+8.08−7.77) % for a lower integration limit of 80 ADC ch.∆a/a =
−16.68+8.37−8.37 % for a lower integration limit of 300 ADC ch.
(5.32)
Strictly speaking, we can only set upper limits on the correction of
the problem in the detector electronics, which are too high to deter-
mine a new value for the neutrino-electron correlation coefficient a,
with a total relative error below the present literature value of 4%
[10], from our latest beam time at the ILL.
128 CHAPTER 5. DATA ANALYSIS
(a) Measurement (b) Simulation
Figure 5.26: (a) Typical (detection) time difference spectra from 18_05_08/morning and
09_05_08/MirrorOff for UA = 50V, with and without electrostatic mirror, respec-
tively. Without electrostatic mirror, only the small fraction of decay protons emitted
with angles 0 ≤ θ0 < 92 ° is detected in coincidence with their correlated electron,
cf. Fig. 5.6b. In comparison with Fig. 5.17b, these spectra stem from measure-
ments at a drift potential of U16A|U16B = −4.2 kV|−0.2 kV. Due to the increased
spatial separation of decay protons and electrons, their peak height is therefore
reduced. (b) Simulated TOF spectra with and without electrostatic mirror, cor-
responding to the data sets 18_05_08/morning and 09_05_08/MirrorOff, respec-
tively. Input data for the MC Simulation (in INM approximation): B0 = 2.177T,
BA/B0 = 0.203, U1 = U1b = 800V, U2 = 1000V, U8R = −50V, U8L = −1000V,
UA = 50V, U16A = −4.2 kV, U16B = −0.2 kV, U17 = −15 kV, number of gener-
ated events = 2 × 106, and a = −0.105 (derived from λ = −1.2701(25) [10]), or,
in the case of measurements without electrostatic mirror: U1 = U1b = U2 ≡ 0 and
U17 = −10 kV. The colorful highlighted areas indicate those areas with poor (red) or
moderate (gray) agreement between the measured time difference and the simulated
TOF spectra. See the text for details. Error bars show statistical errors only.
5.4.2 Backscattering of Decay Electrons Inside the Spectrometer
As we have seen in the previous section (cf. Fig. 5.25), the credibility of our correction for
the saturation effect strongly depends on the simulated proton (and electron) TOF spectra.
Figure 5.26 shows measured time difference spectra in comparison with the expected TOF
spectra from our MC simulations for two different settings of the electrostatic mirror.
As one can see, the simulated TOF spectra show only poor to moderate agreement with
the measured ones. In particular, the measured spectra exhibit coincidence events also
between the actual dead time (5.2µs) of the proton detector and the minimum TOF
(7.1µs) of protons detected in coincidence with the electron from the same decay. Due
to the problem in the detector electronics, by contrast, we would expect a count rate loss
less than 20µs after high-energy electrons (for details see the previous section).
In our measurement, in contrast to the simulation, we cannot decide which proton
belongs to which electron. Therefore, the measured time difference spectra also include
random coincidence events. Figure 5.27a shows the influence of random coincidence events
on the TOF spectra. Obviously, random coincidence events cannot explain the “additional”
coincidence events less than 7.2µs after an electron.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 129
(a) Random coincidences (b) Gaussian broadening
Figure 5.27: Influence of (a) random coincidence events and (b) a Gaussian broadening due to,
e.g. electronic noise on the TOF spectra for UA = 50V. None of the both effects
can explain the deviation from measurement to simulation shown in Fig. 5.26. Input
data for the MC Simulation (in INM approximation): B0 = 2.177T, BA/B0 = 0.203,
U1 = U1b = 800V, U2 = 1000V, U8 = −525V, U16 = −2 kV, U17 = −15 kV, number
of generated events = 108, and a = −0.105 (derived from λ = −1.2701(25) [10]).
Error bars show statistical errors only.
On the other hand, a Gaussian broadening due to, e.g. electronic noise can explain a
minor broadening of the measured time difference spectra, but also not the deviation from
simulation to measurement between 5.2 and 7.2µs, as can be seen from Fig. 5.27b.
In addition, our data analysis has revealed that about
d=ef
Ne
re,p ≈ 20% (5.33)
N0
of the decay electrons (97.45(8) s−1 for the measurement run 16_05_08/night) are de-
tected in our experiment. In principle decay electrons are detected if and only if they
were emitted towards the proton detector and if they can overcome the electromagnetic
mirror right in front of the detector (cf. Fig. 3.8). To be specific: decay electrons, emit-
ted towards the proton detector, with kinetic energies less than 15 keV (cf. Fig 5.18) are
reflected by the detector potential towards the electrostatic mirror, while electrons with
(initial) polar angles of more than 45 ° (see also Fig. 5.28b) are reflected by the magnetic
mirror right in front of the detector. Therefore, we expect that only about11
re,p = 13.1% (5.34)
of the decay electrons are detected in our experiment. This is far too little to explain the
observed fraction Eq. (5.33). We note that the fraction of electrons detected in coincidence
with the proton from the same decay is even smaller due to the spatial separation of decay
electrons and protons.
As stated earlier on page 120 under “Saturation of the Preamplifier”, not more than
1.66% of all decay protons are affected by the saturation effect. Assuming that the full
11Please note that in Refs. [34, 52, 197] the fraction is misstated.
130 CHAPTER 5. DATA ANALYSIS
(a) (b)
Figure 5.28: (a) The values of the magnetic field along the z-axis of the aSPECT spectrometer,
see Fig. 3.8 for details. Potential places where the decay electrons can be reflected
or might be backscattered are additionally highlighted. (b) Angular distribution
of those decay electrons that are not detected by the proton detector at the first
attempt. From the DV towards the bottom flange, the magnetic field drops down
by about a factor of 20. Therefore, from the wire system towards the bottom flange,
the maximum incident angle of the electrons is reduced from about 42° to 15°.
proton decay rate N0 would be reduced by this amount, the observed fraction re,p would
be increased to only about 13.3%. This is not enough to explain the observed fraction
Eq. (5.33).
On the other hand, backscattering of decay electrons inside the aSPECT spectrometer
might explain the observed fraction Eq. (5.33), as shown in the following section. We
mention that, compared to Eq. (5.34), only a fraction of about 85% of all decay electrons
might be backscattered, because of the trapping of decay electrons with (initial) polar
angles 88°< θ0 < 92° between both the magnetic mirror just below the DV and right in
front of the proton detector (for details see Sec. 5.3.4).
Backscattering from the Bottom of the Spectrometer
As explained above, decay electrons not detected at the first attempt were either emitted
towards the electrostatic mirror or reflected towards the electrostatic mirror. Therefore,
it is very likely that decay electrons are backscattered from the bottom of the aSPECT
spectrometer. There, the electrons can be either backscattered from the wire system (e1),
the heat shield(s)12, or the bottom flange of the spectrometer.
Figure 5.28a shows that, from the DV towards the bottom flange of the spectrometer,
the magnetic field drops down by about a factor of 20. For this and geometrical reasons,
about 5% of the decay electrons can hit the wire system, 50% the heat shield, and 50%
12In total, three heat shields were installed between the electrode system and the bottom flange of the
spectrometer, to minimize the heat input from the environment to the cold bore tube of the aSPECT
magnet. The heat shields are arranged one behind the other. For the power supply of the electrodes e1
to e15b as well as the getter pumps (cf. Fig. 3.9a and also Fn. 25 in Chap. 3), all three heat shields have
a cabling hole in their center.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 131
the bottom flange of the spectrometer. Here, the total sum of more than 100% allows for
the possibility that decay electrons are transmitted through the thin (diameter of 125µm)
wires of the wire system. In addition, from the wire system towards the bottom flange,
the maximum incident angle of the decay electrons is reduced from about 42° to 15°, as
can be seen from Fig. 5.28b.
The backscattering of decay electrons from the bottom of the aSPECT spectrometer
was investigated with the program CASINO [227, 228] (see also Fn. 10). As input data
for the CASINO simulations we have chosen:
Wire system: 125µm thin bulk of cooper,
Heat shield(s): composed of
1. 1mm thick substrate of copper,
2. interlayer with 2µm of silver, and
3. surface coating with 1µm of gold.
Bottom flange: > 1mm thick bulk of stainless steel13.
In contrast to SRIM (cf. Fn. 29 in Chap. 3), CASINO allows only the simulation of
one electron energy and incident angle at a time. As can be seen from Fig. 5.29, the
penetration depth and hence also the backscatter probability strongly depend on the
electron incident energy and angle. Therefore, a matrix of electron incident energies and
angles was investigated.
Figure 5.30 shows the results of our CASINO simulations. Obviously, the backscatter
probability varies between 14 and 50%, depending on the incindent energy and angle.
However, only a fraction of the backscattered electrons can pass the magnetic mirror
below the DV, then overcome the electromagnetic mirror right in front of the proton
detector, and finally be detected by the proton detector (cf. Fig. 5.28a).
We mention that our results are in good “agreement” with those presented in Ref. [229]
(cf. also [32]), using the ETRAN [230] electron transport code instead. Like in Ref. [229],
the energy distribution, ρ(X), of backscattered electrons (see Figs. 5.30b, 5.30d, and 5.30f)
could be modeled by(a function: )
− (X − P )
2
ρ(X) = B0 exp , (5.35)
B1(B2 +X)(B3 −X)
where X is the ratio of backscattered energy to incident energy and the parameter P rep-
resents the peak of the distribution. The fit parameters P , B0, B1, B2, and B3 were found
to vary with both incident energy and angle. For the MC simulations of the saturation
effect, we therefore decided to go for a more simplified model, i.e., by linear interpolation
between (X0, ρ0) = (0, 0), (X1, ρ1) = (P, ρ(P )), and (X2, ρ2) = (1, 0).
As stated above, decay electrons can also be transmitted through the thin wires of
the wire system. Although, the corresponding results of our CASINO simulations are not
shown here, they are included in the MC simulations of the saturation effect.
In contrast to more sophisticated programs like, e.g., GEANT4 [231] (see also [232,
233]), CASINO provides no information on the distribution of polar angle, θback, of
13The bottom flange is manufactured from stainless steel AISI-No. 316L (EN-norm 1.4404), i.e., it is
composed of 17% Cr, 12% Ni, 2% Mo, < 0.08% C, < 1% Si, < 2% Mn, < 0.045% P, and < 0.03% S.
132 CHAPTER 5. DATA ANALYSIS
  Agu  31000..00  nm
 Au 0.0 nm   CAguu 0.0 nm
 Au 1000.00 nm
 Ag
 Ag 3000.00 nm
 Cu
4000.0 nm 93000.0 nm
8000.0 nm 186000.0 nm
12000.0 nm 279000.0 nm
16000.0 nm 372000.0 nm
-5746.7 nm -2873.4 nm -0..0  nm 2873.4 nm 5746.7 nm -133611.7 nm-66805.9 nm -0..0  nm 66805.9 nm133611.7 nm
D:\Eigene Dateien\aSPECT\Programs\Casino\heatshield\hs15degrees.cas Position: 0.00nm Energy: 100.00KeV  BackScatteDr:i\nEgig Ceonef fDicaietenite: n0\.a3S9P86E%CT\Programs\Casino\heatshield\hs15degrees.cas Position: 0.00nm Energy: 780.00KeV  BackScattering Coefficient: 0.1724%
(a) Tinc = 100 keV and θinc = 15 ° (b) Tinc = 780 keV and θinc = 15 °
Figure 5.29: CASINO simulations of electron trajectories inside an heat shield, with the upper
edge as incident surface, for an incident angle of 15 ° and two different incident ener-
gies: (a) of 100 keV and (b) of 780 keV. The blue trajectories correspond to electrons
which deposit all energy inside the heat shield, whereas the red ones represent elec-
trons which are backscattered from the heat shield. With increasing incident energy
the penetration depth increases and therefore the backscatter probability decreases.
Please note the different axes ranges in (a) and (b).
backscattered electrons. In Ref. [229], for the case of backscattering from a plastic scintil-
lator, it has been found that this distribution can be well-represented by a parabola with
a peak at 135°:
− (θback − 135°)
2
κ(θback) = 1 , (5.36)2025
where θback is measured with respect to the incident normal, i.e., θback = 180 ° corresponds
to backscatter perpendicular to the surface. If this were the true representation of the
distribution, we would expect that in our experiment a fraction of
rback = 1.3% (5.37)
of all decay electrons is detected additionally to Eq. (5.34). This is too little to explain
the observed fraction Eq. (5.33). Hence, we developed our own model to describe the
distribution:  |θ +θ 2−θ /90°1− back inc| inc |90°+θ |2−θinc/90° for θback > −θincκ(θback) = inc , (5.38)1− |θback+θ |2−θinc/90°inc
|−90°+θ |2−θinc/90°
for θback < −θinc
inc
5.4. UNEXPECTED SYSTEMATIC EFFECTS 133
(a) Wire system (b) Wire system
(c) Heat shield (d) Heat shield
(e) Bottom flange (f) Bottom flange
Figure 5.30: Backscatter probability from the bottom of the aSPECT spectrometer for different
incident energies and angles. (a), (c), and (e) show the backscatter probability from
the wire system, the heat shield(s), and the bottom flange, respectively, as a function
of the electron incident angle, while (b), (d), and (f) present the energy distribution
Eq. (5.35) of backscattered electrons for an incident angle of 15 °. The fit residuals
indicate that our results are in good “agreement” with those presented in Ref. [229].
where θinc denotes the incident polar angle. Here, −90 °< θback < 90 ° is measured with
respect to the z-axis of the spectrometer, i.e., θback = 0 ° corresponds to backscatter
134 CHAPTER 5. DATA ANALYSIS
Figure 5.31: Angular distribution of backscattered electrons from copper for different incident
energies and angles. The green lines correspond to our own model Eq. (5.38), while
the red lines represent a slightly different angular distribution. Namely, a distribution
in which the exponents in Eq. (5.38) have been replaced by 2. The model is based
on measurements of backscattered electrons from copper, silver, and gold [234].
perpendicular to the surface. We note that our model, shown in Fig. 5.31, is based on
measurements of backscattered electrons from aluminum, copper, silver, and gold [234].
Figure 5.32 shows the energy and angular distributions of backscattered electrons from
the bottom of the spectrometer, but now based on our model Eq. (5.38). For this model,
we expect that( in our ex)periment a fraction of
r = 4.24+3.12back −1.52 % (5.39)
of all decay electrons is detected additionally to Eq. (5.34), in moderate agreement with the
observed fraction Eq. (5.33). Here, the error stems from the uncertainty in the polar angle
θback of backscattered electrons. We note that, apart from Eq. (5.38), several different
models were investigated. We further note that our investigation did not include the
possibility that backscattered electrons, not detected at the second attempt, might be
again backscattered from the bottom of the spectrometer, but then into angles θback ≈ 0°
and therefore be detected at the third( attemp)t. In this context it has to be mentioned,that in total about 21.9% of all decay electrons might be reflected from the bottom of the
aSPECT spectrometer, whereas only 7.0+4.3−0.0 % can also overcome the magnetic mirror
below the DV.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 135
(a) (b)
Figure 5.32: (a) Energy and (b) angular distributions of backscattered electrons which are detected
by the proton detector at the second attempt, according to Figs. 5.28b, 5.30, and
Eq. (5.38). From the wire system towards the bottom flange, the magnetic field drops
down (cf. Fig. 5.28a). Therefore, about 1:4:2 electrons are backscattered from the
wire system, the heat shield(s), or the bottom flange of the spectrometer, respectively.
Influence on the Measured Time-of-Flight Spectra
Figure 5.33 shows the measured time difference spectra, previously shown in Fig. 5.26, but
now in comparison with the simulated TOF spectra considering both the backscattering
of decay electrons from the bottom of the aSPECT spectrometer and the saturation of the
preamplifier (for details see page 123 under “Correction for the Saturation Effect”). In the
case of measurements without electrostatic mirror, the backscattering of decay electrons
can resolve the deviation from simulation to measurement completely. In contrast, in
the case of measurements with electrostatic mirror, the backscattering of decay electrons
can only partly explain the deviation from simulation to measurement. We note that, in
the case of measurements without electrostatic mirror, the TOF spectra of backscattered
electrons were scaled down by a multiplication factor of 1/3, in order to fit the measured
time difference spectra. This takes into account the spatial separation of decay protons
and electrons achieved by both, the lower and the upper, dipole electrodes. In the case of
measurements with electrostatic mirror, the TOF spectra of backscattered electrons were
subsequently scaled down by a multiplication factor of only 1/2 (see also [34]), as in these
measurements the upper dipole electrode was used as a cylindrical electrode instead.
As one can see from Fig. 5.6b, TOF differences of less than 9µs can be clearly as-
signed to decay protons emitted towards the electrostatic mirror, with (initial) polar angles
greater than 130 °. Therefore, possible causes for the remaining deviation are:
• proton reflections from the DV due to a charging of the collimation system (investi-
gated in Sec. 5.4.4) or
• a violation of the condition for adiabatic transport due to an enhanced energy trans-
fer from transverse to longitudinal motion (discussed in Sec. 5.4.3).
Since coincidence events additionally to the backscattered electrons did not show up in
measurements without electrostatic mirror, proton reflections from the DV can be ruled
136 CHAPTER 5. DATA ANALYSIS
(a) UA = 50V (b) UA = 250V
(c) Scaled to exponential decay at UA = 50V (d) Scaled to UA = 500V
Figure 5.33: Time difference spectra considering both the backscattering of decay electrons from
the bottom of the aSPECT spectrometer and the saturation of the preamplifier. (a)
and (b) correspond to the data set 09_05_08/MirrorOff, while (c) and (d) correspond
to the data set 18_05_08/morning, see Fig. 5.26 for details. Only in the case of
measurements without electrostatic mirror, the backscattering of decay electrons can
resolve the deviation from simulation to measurement completely. The comparison
between (c) and (d) shows that different scales, because of, e.g., the saturation effect,
can at least minimize the deviation from simulation to measurement. See the text
for details. Error bars show statistical errors only.
out as main explanation for the deviation from simulation to measurement. Hence, we
suggest to decrease the electrostatic mirror potential, in a further beam time.
Correction for the Backscattering of Decay Electrons
From our 3D MC simulations of the saturation effect (for details see page 123 under “Cor-
rection for the Saturation Effect”), we determine a shift in the neutrino-electron correlation
coefficient a due to the backscattering of decay electrons from the bottom of the aSPECT
spectrometer of
∆a/a = (−0.38± 3.46)% (5.40)
where the relative error is dominated by simulation statistics (number of generated
events = 6× 106). We note that, strictly speaking, this is an additional shift in a due to
the saturation effect.
5.4. UNEXPECTED SYSTEMATIC EFFECTS 137
Table 5.4: Proton count rates from 17_05_08/night2 and 18_05_08/night respectively
20_05_08/lunch and 20_05_08/night (after correction for the different magnetic
field ratio, cf. Sec. 5.5.4) for our typical mirror settings U2 = 1000V respectively
U2 = 820V. For comparison, the third, fifth, and seventh column represent the ex-
pected count rates from our MC simulations. We note that the simulated count rates
were normalized to the measured proton count rate for U2 = 820V and UA = 50V.
See the text for details. Input data for the MC simulation (in INM approximation):
B0 = 2.177T, BA/B0 = 0.203, U1 = U1b = 800V, U8R = −50V, U8L = −1000V,
U16A = U16B = −2 kV, U17 = −15 kV, number of generated events = 6× 106 (for each
mirror voltage), and a = −0.105 (derived from λ = −1.2701(25) [10]). The following
abbreviations appear: avg (average value) and CPS (count rate or counts per second).
U2 = 1000V U2 = 820V
avg CPS CPS avg CPS CPS ∆CPS ∆CPS
UA measured simulated measured simulated measured simulated
[V] [s−1] [s−1] [s−1] [s−1] [s−1] [s−1]
50 469.60(33) 467.25(35) 467.15(29) 467.15(35) −2.46(44) −0.10(49)
250 302.31(29) 300.45(28) 300.59(29) 300.34(28) −1.72(41) −0.11(40)
400 151.01(23) 149.91(20) 149.78(21) 149.77(20) −1.23(31) −0.14(28)
500 62.42(15) 62.10(13) 61.63(13) 62.11(13) −0.78(20) 0.01(18)
600 10.43 (9) 10.33 (5) 10.27 (8) 10.32 (5) −0.16(12) −0.01 (7)
5.4.3 Dependence of a on the Electrostatic Mirror Potential
During our latest beam time, the mirror potential U2 was reduced from 1000V to 820V.
The data analysis has revealed that with decreasing mirror potential
• the proton count rates drop down by up to 2.46(44) s−1 for UA = 50V, as can be
seen from Table 5.4, and
• the uncorrected value for the angular correlation coefficient a is reduced by
∆aexp/aexp = (−4.55± 2.63)% (5.41)
from aexp = −0.0978(29) to aexp = −0.1025(28).
Non-Adiabatic Proton Motion ( ( ))
From our MC simulations, we expect far less reduced O 0.1 s−1 proton count rates, as
can be seen from Table 5.4, but also a smaller shift in a of
∆a/a = (−0.55± 3.40)% (5.42)
Here, the error is dominated by simulation statistics (number of generated events = 6×106,
for each mirror voltage). We believe that this shift is due to an enhanced energy transfer
from transverse to longitudinal motion for U2 = 1000V: Decay protons emitted in the
negative z-direction are reflected by the electrostatic mirror at the latest at z = −298.5mm
for U2 = 1000V whereas at z = −318.5mm for U2 = 820V.
138 CHAPTER 5. DATA ANALYSIS
Figure 5.34: Reactor power as measured from the ILL. The colorful highlighted areas correspond
to the measurement runs 17_05_08/night2 and 18_05_08/night with U2 = 1000V
(red) respectively 20_05_08/lunch and 20_05_08/night with U2 = 820V (green).
For different measurement runs, the reactor power changes by ±0.18% (mean value).
However, this is far too little to explain the observed shift Eq. (5.41) or the count
rate losses for U2 = 820V. The latter could be explained by a reduced reactor power
during our measurements with U2 = 820V. As can be seen from Fig. 5.34, a change in
the reactor power of ±0.18% (mean value) for different measurement runs is realistic. A
reduction in that order would correspond to a decrease of the proton count rate of about
1 s−1 for UA = 50V. This is still too little to explain the observed count rate loss of up to
2.46(44) s−1 for UA = 50V.
Penning Discharges in the Bottom of the Spectrometer
On the other hand, it is also possible that the supposed count rate loss for U2 = 820V
is actually a rise in the AP dependent background for U2 = 1000V. The electric field
distribution exhibits a saddle point at the electrostatic mirror. Our electric field calcula-
tions (see also Fig. 3.11 and our note on page 62) have shown that with increasing mirror
potential the saddle point is shifted from z = −364mm to z = −330.5mm. At the same
time, its potential is increased from about 800 to 911.5V, while the potential at height
of the wire system (e1), of at least 782.5V over the entire flux tube, remains unaffected
(see also Fig. 3.11). Hence, for U2 = 1000V, the potential at the saddle point might be
too high to prevent Penning discharges in the bottom part of the aSPECT spectrometer
[235] (see also page 56 under “Penning Traps and Penning Discharge”). For UA ≥ 726V,
only a part of this background contribution can pass the AP, because of the electric retar-
dation method of aSPECT. Thus, neither a background measurement with UA = 780V
(discussed in Sec. 5.2.2) nor measurements with closed neutron shutter (for details see
Sec. 5.3.1) are sufficient to determine such a background contribution. Ultimately, the
proton count rates are larger by this background contribution. Figure 5.35 gives an im-
pression of how such a background contribution could depend on the barrier voltage UA.
In comparison with Fig. 5.11 the observed shift Eq. (5.41) is realistic. Assuming that
the additional background contribution would be as large as shown in Fig. 5.35 (green
5.4. UNEXPECTED SYSTEMATIC EFFECTS 139
Figure 5.35: “Additional” count rate for U2 = 1000V compared to U2 = 820V versus barrier
voltage UA, before (black) and after correction (red) for a reduction of the reactor
power corresponding to Fig. 5.34. For comparison, the green triangles show the
remaining count rates after double correction for the reduction of the reactor power.
We mention that such a reduction reflects the possible influence of a charging of the
collimation system (discussed in the following section). Input data for the correction:
BA/B0 = 0.203 and a = −0.103 [10]. Error bars show statistical errors only.
triangles), the neutrino-electron correlation coefficient a would be shifted by
∆a/a = (+8.50± 4.03)% (5.43)
with a relative error of 47% dominated by counting statistics.
We note that the non-adiabatic proton motion issue can be easily solved by
• electrically decoupling the wire system from its holding electrode (e1b) and subse-
quently
• swapping the mirror voltages U1 and U2, to, e.g., U1 = 800V, U1b = 860V, and
U2 ≤ 400V,
in a further beam time. Our electric field calculations suggest that, then, decay protons
emitted in the negative z-direction are reflected at the latest at z ≤ −371mm. At the same
time, the saddle point is shifted to z ≤ −406mm and its potential is reduced to less than
815.5V. A more sophisticated analysis will be part of further investigations (by means of
MC simulations). However, the question remains whether the observed shift Eq. (5.41)
was mainly due to Penning discharges in the bottom part of the spectrometer. Hence,
we strongly recommend to further investigate this systematic effect, in a further beam
time. In particular, we propose background measurements with different barrier voltages
UA = Uback > Tp,max and measurements with different mirror potentials U1 ∼ Uback. To
obtain meaningful results from these measurements, we highly recommend to improve the
stability of the neutron beam monitor (discussed in Sec. 4.2.1); in order to normalize the
measured proton count rates to the measured neutron count rates.
140 CHAPTER 5. DATA ANALYSIS
5.4.4 Charging of the Collimation System
In addition, our data analysis revealed that for
Measurements in November/December 2007: the proton count rates decreased
with time (cf. Fn. 9 in Chap. 3). We are convinced that the two ceramics rings
(previously used to electrically decouple the long DV cylinders e3 and e6 from the
DV electrode gr) got charged by the decay protons and electrons, what may have
had the effect of proton reflections from the DV (discussed in Sec. 6.4.3).
Measurements with a reduced width of the neutron beam: (see also Sec. 5.5.6)
20mm wide aperture:
• less than 30% of the measurements show a contribution (> 0 counts for
shutter status 5, i.e., when the neutron shutter is closed for 10 s) to back-
ground peak 1, compared to up to 75% for the standard beam width [34].
And less than 15% of the measurements show a contribution to background
peak 2, compared to up to 65% for the standard beam width [34].
• the proton count rate drops down by −11.03(9)% from 471.5(3) s−1
(16_05_08/night) to 419.5(3) s−1 (15_05_08/night), while the electron
count rate remains unaffected (97.52(8) s−1 for the measurement run
15_05_08/night versus 97.45(8) s−1 for 16_05_08/night).
5mm wide aperture: the uncorrected value for the neutrino-electron correlation
coefficient a decreases by
∆aexp/aexp = (−33.93± 7.00)% (5.44)
from aexp = −0.1014(40) (20_04_08/night) to aexp = −0.1358(48)
(16_04_08/night). From our MC simulations of the edge effect (see
Sec. 5.5.6 for details), we expect a shift in a at the level of −10%. But after
considering the saturation of the preamplifier (discussed on page 123 under
“Correction for the Saturation Effect”), we expect a shift in a at the level of
−40%, in agreement with the observed shift Eq. (5.44) (cf. also Fn. 7).
Measurements with a reduced height of the main magnetic field: cf. Sec. 4.2.5
• the slope of the background count rate (cf. Fig. 5.10) increases by a factor of
about 2, for both background peak 1 and peak 2 [34].
• the uncorrected value for the angular correlation coefficient a increases by
∆aexp/aexp = (+31.97± 4.16)% (5.45)
from aexp = −0.1025(28) for B0 = 2.177T to aexp = −0.0697(38) for B0 =
0.933T (21_05_08/night). From our MC simulations (for details see Secs. 5.5.5
and 5.5.6), we expect a shift in a at the level of −5 to −20%. Compared with
the observed shift Eq. (5.45), this is a shift in the wrong direction.
On the basis of our experiences from November/December 2007 and our measurements
with a reduced width of the neutron beam respectively a reduced height of the main
magnetic field, we suspect that the collimation system, shown in Fig. 5.36, got charged
5.4. UNEXPECTED SYSTEMATIC EFFECTS 141
Figure 5.36: A sketch of the collimation system inside the spectrometer. The ruby highlighted
parts show the 6LiF orifices E2, E3, and A1 (cf. Fig. 4.1) and the boron loaded glass
before and behind the decay volume (DV), while the pastel blue rectangular tunnel
shows the DV electrode gr with an opening on one side for vacuum pumping (cf. also
Fn. 8 in Chap. 3). Neutrons are coming from the left and are guided and shaped by
the collimation system through the aSPECT magnet to a beam stop (to the right of
the scheme; not shown here). We suspect that the collimation system got charged
by the decay electrons and protons, cf.Fig. 5.37.
by the decay electrons and protons. We note that with increasing width of the additional
aperture (cf. Sec. 4.1.3) not only the width of the neutron beam inside the DV decreases
by about a factor of 2 or 3 from about 54 to 36 or 18mm, but also the total neutron flux14
decreases by about a factor of 1.5 or 4.5, for the 20 or 5mm wide aperture, respectively.
Similar experiences of the PERKEO II collaboration suggest a charging at the level
of 100V [236]. Figure 5.37 gives an impression of how such a charging of the collimation
system may have changed the electric field distribution inside the DV, which is usually
grounded. For example, we assumed an asymmetrical charging of the collimation system,
i.e., a potential of -150V for the three 6LiF orifices E1, E2, and E3 (see Fig. 4.1) and the
boron loaded glass before the DV, while -50V for the two 6LiF orifices A1 and A2 (see
also Fig. 4.1) and the boron loaded glass behind the DV. This has two consequences:
• the electric field distribution exhibits a saddle point in the DV (see Fig. 5.37a), i.e.,
a local minimum in the y-z-plane (see Fig. 5.37b). Thus decay protons may be
reflected from both the top and the bottom end of the DV electrode gr and finally
trapped inside the DV. We note that, for the bottom half of the neutron beam, also
decay electrons emitted almost perpendicular to the magnetic field may be reflected
from the local electric field minimum in the DV. This, in turn, is at least a partial
explanation for the constant electron count rate independent of the neutron beam
width. Inside the DV the electric field gradient ∂U0/∂z is around −125mVcm−1 for
z < 0 and +125mVcm−1 for z ≥ 0. The latter corresponds to a shift in a of
∆a/a = (+21.82± 1.95)%, (5.46)
14To be precise: The measured neutron flux, assuming that the peak value remains unaffected. For
comparison, in Ref. [34] it was derived that the calculated neutron flux decreases by about a factor of 1.85
in the case of the 20mm wide aperture.
142 CHAPTER 5. DATA ANALYSIS
(a) x-z plane (b) y-z plane
Figure 5.37: Influence of a possible charging of the collimation system on the electric potential in
the decay volume (DV) (a) in the x-z-plane for y = 0 and (b) in the y-z-plane for
x = 0. In our measurement, it is assumed that the DV is grounded, but in this event,
the real potential inside the DV drops down to up to −2V. See the text for details.
Here, the black lines represent the DV electrodes e3, gr, and e6, the 6LiF orifices E3
and A1 (cf. Fig. 4.1), and the boron loaded glass before and behind the DV.
according to Fig. 6.23 in Chap. 6. This is still too little to explain the observed shift
Eq. (5.45). But in addition, the electric field gradient ∂U0/∂z ≈ −125mVcm−1 for
z < 0 can lead to a violation of the condition for adiabatic transport. Therefore,
the angular correlation coefficient a would be shifted to even more positive values.
This could be part of further investigations (by means of MC simulations). We note
that, depending on the real electric field distribution, an electric field parallel to
the neutron beam could help to sweep out decay protons that would otherwise be
trapped.
• the real potential inside the DV drops down from nearly zero to up to −2V. As-
suming an overall decreased electrostatic potential in the DV of U0 = −1V, i.e.,
∆(UA − U0) = +1V, the angular correlation coefficient a would be shifted by
∆a/a = (+12.40± 1.53)%, (5.47)
according to Fig. 3.6. This alone is too little to explain the observed shift Eq. (5.45).
We mention that with a reduced height of the main magnetic field also the drift
perpendicular to both the electric and the magnetic field increases (cf. Eq. (3.39)).
From our MC s{imulations, we expect a drift at the height of the proton detector of
−5 (4) mm , for B0 = 0.933T∆x = { (5.48)−2 (2) mm , for B0 = 2.177T
(−1.5± 4.5) mm , for B0 = 0.933T∆y = (5.49)
0 (2) mm , for B0 = 2.177T
for our standard drift potentials U8R|U8L = −50V|−1000V and U16A = U16B =
−2 kV. We note that the “uncertainties” are dominated by simulation statistics (num-
ber of generated events = 2× 105, for each height of the main magnetic field) and,
for B0 = 0.933T, additionally by the enhanced drift for protons emitted almost per-
pendicular to the magnetic field (cf. Fig. 5.47). With a reduced height of the main
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 143
magnetic field, depending on the real electric field distribution, the volume projected
onto the detector could therefore exhibit an even lower electrostatic potential.
On the other hand, our experiences from November/December 2007 rather suggest a
positive potential for the collimation system. But then, an overall increased electrostatic
potential in the DV of, e.g., U0 = +1V, would yield a shift in a of ∆a/a = (−12.40 ±
1.54)%, in contradiction to the observed shift Eq. (5.45). However, for a charging of the
collimation system, the real electric field distribution inside the DV is hard to predict.
Therefore, we have chosen a more conservative estimate of the error in the neutrino-
electron correlation coefficient a of
∆a/a = (+18.85± 12.4)% (5.50)
Here, the shift stems from proton reflections15 from the DV, while the error stems from
the overall altered electrostatic potential U0, cf. Eq. (5.47).
We note that the charging of the collimation system issue can be easily solved by
surface coating with, e.g., aluminum or titanium, in a further beam time. However, the
work function differences both between the collimation system and the proposed surface
coating as well as between the surface coating and the gold-plated surfaces of our electrode
system should still be investigated (cf. also Chap. 6). Hence, we strongly recommend to
repeat both the tests with a reduced width of the neutron beam and with a reduced height
of the main magnetic field in a further beam time. In particular, we propose measurements
with several different widths of the neutron beam profile and at least three different heights
of the main magnetic field. Given the changes in the mirror potentials proposed in the
previous section, we also suggest to re-examine the influence of non-adiabatic proton
motion (discussed in Sec. 3.4.2) on a (by means of MC simulations).
5.5 Investigations of Systematic Effects
During our latest beam time, several different systematic effects were investigated exper-
imentally (see also Sec. 4.2.5). Here, we present the results of their analysis, primarily
with regard to their impact on the neutrino-electron correlation coefficient a.
5.5.1 Electrostatic Mirror Potential
As mentioned earlier in Sec. 3.1 (see also page 57 under “The Electrostatic Mirror”), the
electrostatic mirror is held at a positive voltage UM > Tp,max, in order to reflect all
decay protons emitted in the negative z-direction. To verify that our two typical voltage
settings (see Table 3.1) of the electrostatic mirror electrodes e1, e1b, and e2 are sufficient
for this purpose, the mirror potential was reduced in several steps from U1(= U1b)|U2 =
1080V|1100V to 0V|0V.
Our electric field calculations (see also Fig. 3.11 and our note on page 62) have shown
that a mirror potential of U1|U2 = 760V|780V already ensures a rather homogeneous
electric field distribution of at least 760V (740V at the height of the wire system e1)
over the entire flux tube. The measurements have confirmed that a potential of U1|U2 =
760V|780V guarantees 100% acceptance for decay protons, as can be seen from Table 5.5.
15Please note that mostly the top half of the neutron beam is affected by the proton reflections and
therefore the angular correlation coefficient a is only shifted by ∆a/a = (+18.85 ± 1.63)%, compared to
Eq. (5.46).
144 CHAPTER 5. DATA ANALYSIS
Table 5.5: Proton count rates from 14_05_08/mirror and 14_05_08/mirror_rate for different
settings of the electrostatic mirror electrodes e1, e1b, and e2, for UA = 50V. A potential
of U1|U2 = 760V|780V already ensures that all decay protons emitted in the negative
z-direction will be reflected by the electrostatic mirror. In comparison with the top
of Fig. 5.9, these count rates stem from measurements at an acceleration potential of
−10 kV and are therefore lower by about 15 to 20Hz.
U1 = U1b U2 Proton count rate Comment
[V] [V] [s−1]
1080 1100 452.40(92)
800 1000 449.98(20) typical voltage setting
850 900 450.60(92)
800 820 not measured typical voltage setting
760 780 451.10(92)
0 0 228.58(12)
Given the changes in the mirror potentials proposed in Sec. 5.4.3, we recommend to
repeat this measurement in a further beam time.
5.5.2 Magnetic Mirror Effect in the Decay Volume
As discussed earlier in Sec. 5.3.4, a small fraction, δ, of all decay protons, emitted in the
negative z-direction, is reflected by the local magnetic field maximum just below the DV
(see Fig. 4.5b). From the measurements both of the neutron beam (presented in Sec. 4.1.3)
and the magnetic field profiles (see Sec. 4.1.4 for details), this fraction can be calculated.
Count Rate Ratio With and Without Electrostatic Mirror
In our experiment, the fraction δ cannot be measured directly, but indirectly via the proton
count rates with and without electrostatic mirror:
Np(UA = 0V;UM = 0V) N0(0.5 + δ)= = 0.5 + δ . (5.51)
Np(UA = 0V;UM > Tp,max) N0
In the first order, the count rate ratio Eq. (5.51) does not depend on the drift potentials
U8R|U8L and U16A|U16B. Hence, the count rate ratio can be calculated by computing the
proton’s trajectories only in (2D) axially symmetric electric and magnetic fields (see also
our note on page 62). From our MC {simulations, we expect a count rate ratio of
N (U ;U = U = 0V) 51.052± 0.004(stat)+0.018p A 1 2 −0.006(sys) % , for UA = 0V=
Np(U ;U1|U2 = 800V|1000V) 51.048± 0.003(stat)+0.018A −0.005(sys) % , for UA = 50V
(5.52)
Here, the systematic error stems from the poor quality of the simulated beam profiles
(discussed in Sec. 4.1.3) and the uncertainty in the exact position (including the rotation
angle in the x-y-plane) of the proton detector relative to the neutron beam (determined
in Sec. 5.5.6), cf. Fig. 5.38a.
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 145
(a) (b)
Figure 5.38: (a) Proton count rate ratio with and without electrostatic mirror for different posi-
tions of the proton detector (for details see Sec. 5.5.6). We note that the neutron beam
profile varies only marginally over the depth of the flux tube (cf. Sec. 4.1.3). Input
data for the MC simulation (in INM approximation): B0 = 2.177T, BA/B0 = 0.203,
U1 = U1b = 800V, U2 = 1000V, U8 = −525V, UA = 50V, U16 = −2.2 kV,
U17 = −10 kV, number of generated events = 108, and a = −0.105 (derived
from λ = −1.2701(25) [10]). Except for the settings of the lower dipole electrode
(e8), the input data correspond to the data set 14_05_08/mirror_rate. The error
bars show statistical errors only and are strongly correlated, as each point stems
from the same MC simulation. (b) Background subtracted pulse height spectra
from 14_05_08/mirror_rate with (red) and without (blue) electrostatic mirror for
UA = 50V. In comparison with Fig. 5.8b, both spectra stem from measurements at
an acceleration potential of −10 kV and therefore the proton peak is shifted to lower
ADC channels (see also Fig. 5.3). The error bars show statistical errors only.
To investigate the alignment between the magnetic field and the neutron beam, the
proton count rates with and without electrostatic mirror were compared. From the mea-
surements, shown in Fig. 5.38b (see also Table 5.5), we derive a count rate ratio of
Np(50V;U1 = U2 = 0V) = 50.80(3)%. (5.53)
Np(50V;U1|U2 = 800V|1000V)
The result is 5.2 standard deviations from our expectation Eq. (5.52). According to Fn. 8
in Chap. 4, a slightly different magnetic field profile (see Fig. 4.5b) might at least partly
explain this deviation, but would rather suggest an even higher count rate ratio.
With increasing barrier voltage UA the slope of the transmission function Eq. (3.30)
increases. Therefore, less protons with polar emission angles around 90 degrees can pass
the AP (see also Fig. 3.2). Ultimately, the count rate ratio decreases down to nearly
50% for UA = 600V, as can be seen in Fig. 5.39 (green triangles). Unfortunately, the
measured count rate ratios are in poor agreement with the predicted values from our
MC simulations. Even computing the proton’s trajectories in (3D) non-axially symmetric
electric fields cannot improve the poor agreement, as can also be seen from Fig. 5.39. We
are convinced that this is attributed to non-statistically fluctuating count rates during
these measurements (cf. Fn. 7 and also Fig. 5.15).
146 CHAPTER 5. DATA ANALYSIS
Figure 5.39: Proton count rate ratio with and without (count rates from 09_05_08/MirrorOff)
electrostatic mirror versus barrier voltage UA. The black squares and red circles stem
from measurements performed at a time when the count rates were fluctuating non-
statistically (cf. Fn. 7). For comparison, the triangles show the expected ratios from
our MC simulations, where the blue ones also consider the drift potentials of both, the
lower and the upper, dipole electrodes. Input data for the MC simulations (in INM
approximation): B0 = 2.177T, BA/B0 = 0.203, U1 = U1b = 800V, U2 = 1000V,
U8R = −50V, U8L = −1000V, U16A = −4.2V, U16B = −0.2 kV, U17 = −10 kV,
number of generated events = 6×106 (for each AP voltage), and a = −0.105 (derived
from λ = −1.2701(25) [10]); or, in the case of our computations in (2D) axially
symmetric electric fields: U8 = −525V, U16 = −2.2 kV, and number of generated
events = 108 (for each AP voltage). Error bars show statistical errors only.
Conclusions for a Charging of the Collimation System
Let us assume that the full proton decay rate N0 is reduced by a factor x while the
proton count rate without electrostatic mirror is only reduced by a factor y = t · x, with
a parameter 0 ≤ t ≤ 1. Then, the count rate ratio for UA = 50V changes to about
0.5106 · (0.9619 ·N0)− y · (0.9619 ·N0) 0.5106− t · x= =! 0.508. (5.54)
(0.9619 ·N0)− x · (0.9619 ·N0) 1− x
Solving Eq. (5.54) for x yields:
0.0026 !
x = > 0, (5.55)
t− 0.508
and therefore t > 50.8%. However, during our latest beam time, we observed the effects
of the saturation of the preamplifier (discussed on page 120) only less than 20µs after
high-energy electrons. According to Fig. 5.6, the saturation effect can therefore be ruled
out as main explanation for the deviation from simulation to measurement.
On the other hand, a charging of the collimation system (discussed in Sec. 5.4.4) can
lead to unexpected proton reflections from the DV (see also Sec. 6.4.3). The protons
affected may be trapped between the DV and the electrostatic mirror, what, in turn, will
lead to a reduction of the proton count rate. This mostly affects the protons emitted almost
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 147
Figure 5.40: Proton count rates from 20_04_08/trans for different upper dipole voltages U16A =
U16B. For better visibility, the y-axis is broken at 353 s−1. Because of the lack of back-
ground measurements with UA = 780V, the background count rate was estimated
to be 3.5 s−1. In comparison with the top of Fig. 5.8b, these count rates stem from
measurements at an acceleration potential of −10 kV and, additionally, an increased
length of trigger window w2 of 25 time bins, and are therefore lower by about 30Hz.
For comparison, the red circles show the expected count rates from our MC simula-
tions. For |U16A| < 0.6 kV, the measured proton count rates do not fit the simulated
count rates (cf. also Fn. 7), even after taking account of the saturation of the pream-
plifier (green triangles). We note that the simulated count rates were normalized to
the measured proton count rates for |U16A| ≥ 0.6 kV. Input data for the MC simu-
lations (in INM approximation): B0 = 2.177T, BA/B0 = 0.203, U1 = U1b = 800V,
U2 = 1000V, U8R = −50V, U8L = −1000V, UA = 50V, U17 = −10 kV, number of
generated events = 5× 105 (for each dipole voltage), and a = −0.105 (derived from
λ = −1.2701(25) [10]). Error bars show statistical errors only.
perpendicular to the magnetic field (cf. Figs. 3.2 and 6.22), what speaks for t  50.8%.
We mention that solving Eq. (5.54) for y, in turn, yields
!
y = 0.0026 + 0.508 · x ≤ x, (5.56)
and hence
x ≥ 0.53% (5.57)
In comparison with Table 5.3 this is quite realistic.
In summary, measurements with and without electrostatic mirror have to be repeated
in a further beam time. Again (cf. Sec. 5.4.3), we recommend to improve the stability of
the neutron beam monitor.
5.5.3 Magnetic Mirror Effect in Front of the Proton Detector
As mentioned earlier on page 59, the upper dipole electrode e16 also ensures that all decay
protons that pass the potential barrier can overcome the magnetic mirror right in front
148 CHAPTER 5. DATA ANALYSIS
of the proton detector (see Fig. 4.5a). To verify that our typical voltage settings (see also
Table 3.1) of the upper dipole electrodes e16A and e16B are sufficient for this purpose,
the dipole potential was enlarged in several steps from U16A = U16B = −200V to -2000V.
Our electric field calculations (see Fig. 3.8 and our note on page 62) have shown that a
dipole potential of U16A ≈ −600V already guarantees 100% acceptance of decay protons.
Our measurements have confirmed that a potential of U16A = −600V ensures that all
decay protons concerned also overcome the magnetic mirror in front of the detector, as
can be seen from Fig. 5.40.
But for |U16A| < 600V, the measured proton count rates do not match the simulated
count rates, even after considering the saturation of the preamplifier (discussed on page 120
under “Correction for the Saturation Effect”). We note that these measurements were
performed at a time when the count rates were fluctuating non-statistically (cf. Fn. 7). It
is therefore very difficult to decide what is the major cause of
• increased proton count rates for |U16A| < 600V or possibly
• reduced proton count rates for |U16A| ≥ 600V.
To summarize, possible causes include:
• non-statistically fluctuating count rates (see Fig. 5.15),
• the increased length of trigger window w2, from usually 24 to 25 time bins (see
page 69 under “Digital Electronics and the Trigger Algorithm”). The comparison
with Table 5.5 shows that a loss rate at the level of 10Hz for UA = 50V is realistic.
• the saturation of the preamplifier; owing to Fig. 5.40 this is almost excluded,
• an enhanced energy transfer from transverse to longitudinal motion for U2 = 1000V
(discussed on page 137 under “Non-Adiabatic Proton Motion”). This mainly affects
the protons with large emission angles to the magnetic field axis (cf. Fn. 3 in
Chap. 3). Hence, it will lead to an increase of the proton count rates for |U16A| <
600V, while the proton count rates for |U16A| ≥ 600V remain unaffected.
• a possible charging of the collimation system. According to the previous section,
this mostly affects the protons emitted almost perpendicular to the magnetic field.
Therefore, it will lead to a reduction of the proton count rates for |U16A| ≥ 600V,
while leave the proton count rates for |U16A| < 600V nearly unaffected.
Hence, for measurements with |U16A| ≤ 1000V and |U16B| ≤ 1000V, we propose to repeat
this test in a further beam time.
5.5.4 Ratio of the Magnetic Fields
To test the calculated transmission function Eq. (3.30), the ratio of the magnetic fields in
the AP and the DV, rB = BA/B0, was changed (see Secs. 3.2.2 and 4.2.5 for details) by
∆rB/rB = 0.94(21)% (5.58)
from the standard ratio rB = 0.2030(3) to rB = 0.2049(3) [53]. Unfortunately, with a
relative error of 22% dominated by the accuracy of the Hall probe (discussed in Sec. 4.1.4).
From our simulations, we expect a shift in the angular correlation coefficient a of
∆a/a = (−8.65± 1.46+1.41−1.39)%, (5.59)
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 149
if Eq. (3.32) would be fitted to the integral proton spectrum by using the standard ratio
rB = 0.203. Here, the first error stems from fitting Eq. (3.32) to the simulated proton
count rates and the latter one from the uncertainty in the magnetic field ratios. From the
measured proton{count rates (20_05_08/night), we determine a value for a of
−0.1029(34) , for Ftr(T ;UA, rB = 0.2049) in Eq. (3.32)aexp = (5.60)
−0.1141(34) , for Ftr(T ;UA, rB = 0.203 ) in Eq. (3.32)
This corresponds to a shift in the neutrino-electron correlation coefficient a of
∆aexp/aexp = (−10.89± 4.93)%, (5.61)
in agreement with the expected value Eq. (5.59) from our simulations. We note that
another measurement at the same day (20_05_08/lunch), but with the standard ratio
rB = 0.203 instead, leads to aexp = −0.1020(45). This is also consistent with Eq. (5.60).
But obviously our statistics was too low to draw further conclusions concerning an
angular-dependent detection efficiency of the aSPECT spectrometer:
Let us assume that protons emitted almost perpendicular to the magnetic field will
either not reach the proton detector or will not be counted by the detector. To
be specific: The proton count rate is reduced by protons with an (initial) polar
angle 90°≥ θ0 > θmaxacc ∼ 90° (cf. also Fn. 3 in Chap. 3), where θmaxacc denotes the
maximum polar angle accepted. With increasing barrier voltage UA the slope of
the transmission function Eq. (3.30) increases and therefore the relative count rate
loss decreases (see also Fig. 3.4). Thus, with decreasing maximum angle accepted
θmaxacc the angular correlation coefficient a is shifted to more positive values, as can be
seen in Fig. 5.41. This shift slightly depends on the ratio of the magnetic fields rB =
BA/B0, as can also be seen from Fig. 5.41. Hence, high-precision measurements with
different magnetic field ratios might reveal a possible angular-dependent detection
efficiency of our spectrometer.
Hence, we suggest to repeat this measurement in a future (high-precision) beam time; if
possible, with different and more radically changed magnetic field ratios. We recommend
at the same time to improve the stability of the neutron beam monitor (see also Sec. 5.4.3).
5.5.5 Height of the Main Magnetic Field
To study the influence of the non-adiabatic proton motion (discussed in Sec. 3.4.2; see
also Sec. 5.4.3) on the neutrino-electron correlation coefficient a, measurements with two
different heights of the main magnetic field were performed. As mentioned earlier in
Sec. 5.4.4, with decreasing main magnetic field the uncorrected value for the angular
correlation coefficient a increases by
∆aexp/aexp = (+31.97± 4.16)% (5.62)
from aexp = −0.1025(28) for B0 = 2.177T to aexp = −0.0697(38) for B0 = 0.933T
(21_05_08/night). From the MC simulations [47], we expect a much smaller shift in a
of16
0.5% ∆a/a < 4%, (5.63)
16Please note that in Refs. [34, 53] the shift in a was greatly underestimated with |∆a/a| > 0.4%.
150 CHAPTER 5. DATA ANALYSIS
Figure 5.41: Top: Relative change of the angular correlation coefficient a for different maximum
polar angles accepted θmaxacc (for details see the text). With decreasing maximum
angle accepted θmaxacc the angular correlation coefficient a is shifted to more positive
values. The comparison between the black squares (BA/B0 = 0.203) and the red
circles (BA/B0 = 0.2049) shows that the shift slightly depends on the ratio of the
magnetic fields rB = BA/B0. Bottom: For elucidation, the green squares show the
difference between the red circles and the black squares. Input data for the simulation:
UA = 50, 250, 400, 500, 600V and the recommended value for a = −0.103 [10].
cf. Table 3.2. This is far too little to explain the observed shift Eq. (5.62).
On the other hand, with decreasing main magnetic field the proton’s radius of gyration
increases (cf. Eq. (3.3)) and hence both transition regions shown in Fig. 3.23 get wider.
Therefore, the impact of the edge effect (described in Sec. 3.4.6) on the angular correlation
coefficient a increases. However, from our MC simulations of the edge effect (see the
following section), we expect a shift in a of
∆a/a = −22.82(97) %, (5.64)
cf. Table 5.7. Compared with the observed shift Eq. (5.62), this is a shift in the wrong
direction.
Even computing the proton’s trajectories in (3D) non-axially symmetric electric fields
cannot improve the poor agreement, but at least reveal a possible underestimation of the
change in a due to non-adiabatic proton motion. From our 3D MC simulations, we expect
a shift in a of only
∆a/a = (−7.03± 3.83) %. (5.65)
We note that with decreasing main magnetic field the proton’s drift increases (cf.
Eqs. (5.48) and (5.49); see also Table 5.6) and hence the spatial separation of decay
protons and electrons on the proton detector increases. Therefore, the saturation effect of
the detector electronics (discussed on page 120 under “Saturation of the Preamplifier”) is
reduced. Altogether, the comparison with our calculations in 2D axially symmetric fields
Eq. (5.64) results in a shift in the angular correlation coefficient a due to non-adiabatic
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 151
proton motion of
∆a/a = (+15.79± 3.96)% (5.66)
with a relative error of 25% dominated by simulation statistics (number of generated
events = 6 × 106 for B0 = 2.177T and 12 × 106 for B0 = 0.933T). Compared with
Eq. (5.63), this is much higher than expected [47].
We mention that the figures in Table 4.1 (from Ref. [47]) are based on a potential
difference of |U8R − U8L| = 3 kV (see also Fn. 14 in Chap. 3). Thus, it is unclear whether
the expected shift in a from Ref. [47] for B0 = 2.177T of
0.001% < ∆a/a < 0.04% (5.67)
also underestimates the actual impact of non-adiabatic proton motion on a. Hence, we
strongly recommend to re-examine the influence of non-adiabatic proton motion on a (by
means of MC simulations). In addition, we suggest to repeat this test with at least three
different heights of the main magnetic field, in a further beam time.
5.5.6 Edge Effect
To determine the correction for the edge effect (introduced in Sec. 3.4.6), measurements
with three different widths of the neutron beam profile (see also Secs. 4.1.3 and 4.2.5)
were performed. As explained earlier (in Sec. 3.4.6), the edge effect strongly depends
on the shape of the neutron beam profile and hence also on the exact position of the
proton detector relative to the rest of the electrode system and the magnetic field. The
high voltage electrode e17 (surrounding the detector) is mounted to a long arm17, cf.
Fig. 3.20a. Therefore, a deflection of the long arm out of the magnetic field axis by a few
tenth of a degree can lead to a displacement of the proton detector in the x-y-plane of
several millimeters. We note that a deflection of the superconducting coil system (fixed
relative to the DV) out of the symmetry axis of the electrode system has the same effect.
This is not considered in the present thesis, but could be part of further investigations (by
means of MC simulations). Consequently, for each width of the neutron beam profile, we
first aligned the “center” of the neutron beam on the detector. Therefore, we compared
the proton count rates for several different settings of both the lower and the upper dipole
electrodes e8 and e16, respectively.
The rotation angle of the proton detector in the x-y-plane, φdet, was measured to be
φdet = 5.7(5) ° (5.68)
Fig. 5.42 suggests that this rotation corresponds to a widening of the projected area (see
Fig. 3.23), by about 10% (cf. Fig. 5.45). Actually, the figure also shows how a shift of
−3mm in both the x- and the y-direction would look like.
Determination of the Position of the Proton Detector
According to Eq. (3.39) (see also Fig. 5.47)
• the lower dipole electrode e8 mainly18 causes a drift of the decay protons in x-
direction, i.e., parallel to the neutron beam, whereas
17The distance between the center of rotation and the proton detector is about 1.4 meters.
18To be precise, the dipole electrodes cause a transvection (shear mapping) of the flux tube, as can also
be seen from Fig. 5.47.
152 CHAPTER 5. DATA ANALYSIS
(a) x = y = 0 and φdet = 0 ° (b) x = y = −3mm and φdet = 5.7 °
Figure 5.42: A sketch of the electromagnetic set-up, see Fig. 3.18b for details. The comparison
between (a) and (b) shows the effect of both a rotation and a shift of the proton
detector by, e.g., ∆φdet = 5.7 ° and ∆xdet = ∆ydet = −3mm, respectively.
• the upper dipole electrode e16 mainly (cf. Fn. 18) causes a drift in y-direction, i.e.,
perpendicular to the neutron beam.
Therefore, the displacement of the proton detector in x- and y-direction was determined
separately by measurements with different lower and upper dipole voltages, respectively.
From the measurements with an additional aperture, shown in Fig. 5.43, we derive a
displacement of the proton detector in y-direction of
∆ydet = −3.5(5)mm (5.69)
The same result was obtained for a measurement with the 20mm wide aperture but
UA = 400V.
We would like to emphasize that our data analysis revealed that the poor quality
(cf. Figs. 4.4 and 4.3) of the simulated neutron beam profiles (from Ref. [34]) was not
adequate to describe the measured proton count rates. Therefore, the simulated neutron
beam profiles were smoothed by bilinear interpolation (see also Sec. 4.1.3). In particular,
for the full width of the neutron beam, the simulated neutron beam profile results in a
displacement of only ∆ydet = −1.5(5)mm, even after smoothing of the beam profile, as
can be seen from Figs. 5.44a and 5.44b. Even though the input parameters for the MC
simulation were adapted to the measured beam profiles (cf. Ref. [34]), this deviation from
Eq. (5.69) cannot be explained by the overall spatial accuracy of our measured beam
profiles of 1mm (for details see Sec. 4.1.3). Hence, for the full width of the neutron
beam, the neutron beam profile in the DV was determined by bilinear interpolation of the
measured beam profiles in front of the entrance window (see Fig. 4.3a) and behind the
exit window (see Fig. 4.3c) of the aSPECT magnet instead. This yields a displacement
of ∆ydet = −3.0(5)mm, in agreement with Eq. (5.69). For comparison, the extrapolation
of the measured beam profile behind the exit window, shown in Fig. 5.44c, results in a
displacement of ∆ydet = −3.5(5)mm, as can be seen from Fig. 5.44d. For our analysis,
we therefore use the extrapolated neutron beam profile. We note that the saturation of
the preamplifier (discussed on page 120) has only minor impact on Figs. 5.43 to 5.46.
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 153
(a) 5mm wide aperture (b) 20mm wide aperture
Figure 5.43: Proton count rates (a) from 17_04_08/uExB with the 5mm wide aperture and (b)
from 16_05_08/uExB2 with the 20mm wide aperture for different upper dipole
drifts U16A|U16B. The colored circles show the expected count rates from our MC
simulations for different positions of the proton detector. Input data for the MC
simulations (in INM approximation): B0 = 2.177T, BA/B0 = 0.203, U1 = U1b =
800V, U2 = 1000V, U8R = −50V, U8L = −1000V, UA = 50V, U17 = −10 kV,
number of generated events = 5 × 105 (for each dipole voltage), and a = −0.105
(derived from λ = −1.2701(25) [10]), or, in the case of measurements with the 20mm
wide aperture: U1 = U1b = 800V, U2 = 820V, U8R = 0V, U8L = −200V, UA = 50V,
U17 = −15 kV. Error bars show statistical errors only.
As mentioned above, the rotation angle φdet of the proton detector was measured. In
the case of Figs. 5.43, 5.44, and also 5.46, this has no influence on the determination of the
position of the proton detector. In contrast to a measurement with a reduced height of the
main magnetic field, as can be seen from Fig. 5.45. This measurement yields a displacement
of ∆ydet = −3.0(5)mm, also in agreement with Eq. (5.69) (cf. also Sec. 5.4.4). Taken
together, we determine the displacement of the proton detector in y-direction to
∆ydet = (−3.5± 1.0)mm (5.70)
As mentioned earlier in Sec. 4.1.3, the neutron beam profile varies only slightly over
the depth, x, of the flux tube. But, with increasing drift potential U8R|U8L the left and/or
the right detector pad will step out from or into the shadow of the electrode system.
Hence, the displacement of the detector in x-direction can only be determined from the
proton count rates in an outer detector pad, i.e., in detector channel 5 or 7. From the
measurements with several different settings of the lower dipole electrode e8, shown in
Fig. 5.46, we derive a displacement of the proton detector in x-direction of
∆xdet = −3(1)mm (5.71)
Here, the uncertainty also considers that these measurements were performed at a time
when the count rates were fluctuating non-statistically (cf. Fn. 7).
154 CHAPTER 5. DATA ANALYSIS
(a) Simulated neutron beam profile (b) Simulated neutron beam profile
(c) Profile measured behind the exit window (d) Profile measured behind the exit window
Figure 5.44: Uncertainty in the position of the proton detector due to the knowledge of the
neutron beam profile, (a) and (b) for the simulated beam profile and (c) and (d)
for the neutron beam profile measured behind the exit flange. (a) and (c) show
the neutron beam profiles, while (b) and (d) show the proton count rates from
17_05_08/uExB_fullbeam for different upper dipole drifts U16A|U16B. The col-
ored circles show the expected count rates from our MC simulations for different
positions of the proton detector. See the text for details. Input data for the MC sim-
ulations (in INM approximation): B0 = 2.177T, BA/B0 = 0.203, U1 = U1b = 800V,
U2 = 820V, U8R = 0V, U8L = −200V, UA = 50V, U17 = −15 kV, number of gen-
erated events = 5 × 105 (for each dipole voltage), and a = −0.105 (derived from
λ = −1.2701(25) [10]). Error bars show statistical errors only.
Dependence of the Drift on the Proton’s Momentum
In addition to the drift potentials U8R|U8L and U16A|U16B, the proton’s drift also depends
on its flight time through the dipole electrodes. The flight time, in turn, depends on the
proton’s velocity component v|| parallel to the magnetic field. Therefore, the proton’s drift
caused by the dipole electrodes e8 and e16 depends on both the proton (initial) kinetic
energy, T0, and its polar angle, θ0. However, in the case of the upper dipole electrode,
the decay protons are additionally post-accelerated by the high negative potential of the
detector electrode e17. For, e.g., U16A = U16B = −2 kV and U17 = −15 kV, the acceleration
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 155
(a) B0 = 0.933T, φdet = 0 ° (b) B0 = 0.933T, φdet = 5.7 °
Figure 5.45: Uncertainty in the position of the proton detector due to its rotation angle, (a) for
φdet = 0 ° and (b) for φdet = 5.7 °. (a) and (b) show the proton count rates from
22_05_08/uExB for different upper dipole drifts U16A|U16B. The colored circles
show the expected count rates from our MC simulations for different positions of the
proton detector. See the text for details. Input data for the MC simulations (in INM
approximation): B0 = 0.933T, BA/B0 = 0.203, U1 = U1b = 800V, U2 = 820V,
U8R = −50V, U8L = −1000V, UA = 50V, U17 = −15 kV, number of generated
events = 5 × 105 (for each dipole voltage), and a = −0.105 (derived from λ =
−1.2701(25) [10]). Error bars show statistical errors only.
(a) Detector channel 5 (b) Detector channel 6
Figure 5.46: Proton count rates from 07_05_08/lExB for different lower dipole drifts U8R|U8L,
(a) for an outer detector pad and (b) for the central detector pad. The count rates
< 20 s−1 correspond to measurements with positive voltages U8R, U8L > 0. Thus,
for better visibility, the y-axis is broken at 21 s−1. The colored circles show the
expected count rates from our MC simulations for different positions of the proton
detector. Input data for the MC simulations (in INM approximation): B0 = 2.177T,
BA/B0 = 0.203, U1 = U1b = 800V, U2 = 1000V, UA = 50V, U16A = −4.2 kV,
U16B = −0.2 kV, U17 = −10 kV, number of generated events = 5 × 105 (for each
dipole voltage), and a = −0.105 (derived from λ = −1.2701(25) [10]). Error bars
show statistical errors only.
156 CHAPTER 5. DATA ANALYSIS
potential drops from about −1.35 kV at the bottom edge to −6.7 kV at the top edge of
the upper dipole electrode. In addition, the magnetic field at height of the upper dipole
electrode is higher by about a factor of 1.8 compared to the lower dipole electrode and
hence the drift velocity Eq. (3.39) is reduced by about a factor of 0.55. Therefore, in
the case of the upper dipole electrode, we expect a suppressed angular and a strongly
suppressed energy dependence of the proton’s drift.
Figure 5.47 shows the dependence of the drift on both the proton kinetic energy and
its polar angle for two different settings of the aSPECT spectrometer. Obviously, the drift
is enhanced for both protons emitted with low kinetic energy and/or almost perpendicular
to the magnetic field. Moreover, the figure confirms our expectation that, in the case of
the upper dipole electrode, both the energy and the angular dependence are suppressed.
However, compared to Eq. (5.70), both the energy and the angular dependence of the
proton’s drift are in the same order of magnitude as the uncertainty in the position of
the proton detector. In view of the low simulation statistics in the case of computing the
proton’s trajectories in (3D) non-axially symmetric electric fields, we go for calculations in
(2D) axially symmetric electric and magnetic fields (see also Sec. 5.5.2). Then, additional
3D MC simulations serve to
• determine the mean proton drift in dependence of the settings of the spectrometer,
in order to correct the calculations in 2D axially symmetric fields for the proton’s
drift, and
• cross-check our combined simulations (i.e., compute the proton’s trajectories in 2D
axially symmetric fields and “add” the mean proton drift).
From our 3D MC simulations, we determine the mean proton drifts presented in Table 5.6.
For a precision measurement of the neutrino-electron correlation coefficient a, i.e,
a measurement with at least optimized pulse shaping and reduced amplification of the
preamplifier (cf. Sec. 5.4.1) in a further beam time, we strongly recommend to addition-
ally integrate the slight energy and angular dependencies (see Fig. 5.47) of the proton drift
into the combined simulations. This, however, requires improved determinations of the
neutron beam profile(s) and also the position of the proton detector relative to the rest of
the electrode system and the magnetic field.
Dependence of a on the Position of the Proton Detector
As explained above, calculations in 2D axially symmetric fields serve to determine the
influence of the edge effect on the neutrino-electron correlation coefficient a. Figure 5.48
shows the results of our 2D MC simulations for measurements with the standard settings
of the aSPECT spectrometer, reduced widths of the neutron beam as well as a reduced
height of the main magnetic field.
The comparison between Figs. 5.48a to 5.48c shows the strong dependence of the edge
effect on the shape of the neutron beam profile. With increasing width of the neutron
beam, the impact of the edge effect on the angular correlation coefficient a decreases: The
strong dependence of the shift in a on the exact position of the proton detector drops down
from ∆a/a ∈ (−10.5%,+2.5%) for the 5mm wide aperture to ∆a/a ∈ (−2.2%,+1%) for
the full width of the neutron beam. Thus, also the magnitude of the shift in a decreases.
In addition, the comparison between Figs. 5.48c and 5.48d shows the dependence of
the edge effect on the width of both transition regions (see Fig. 3.23). With decreasing
height of the main magnetic field the proton’s radius of gyration Eq. (3.3) increases and
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 157
(a) B0 = 2.177T (b) B0 = 2.177T
(c) B0 = 0.933T (d) B0 = 0.933T
Figure 5.47: Drift of the decay protons in the x-y-plane for different settings of the aSPECT
spectrometer: (a) and (b) for Imain = 70A and U16A|U16B = −4.2 kV|−0.2 kV, and
(c) and (d) for Imain = 30A and U16A = U16B = −2 kV. (a) and (c) show the energy
dependence of the proton drift, while (b) and (d) present the angular dependence. We
note that the lower dipole electrode e8 mainly causes a drift of the decay protons in
x-direction, while the upper dipole electrode e16 causes a drift in y-direction. Further
input data for the MC simulations (in INM approximation): BA/B0 = 0.203, U1 =
U1b = 800V, U2 = 820V, U8R|U8L = −50V|−1000V, UA = 50V, U17 = −15 kV,
number of generated events = 2 × 106 (for each setting), and a = −0.105 (derived
from λ = −1.2701(25) [10]). For better visibility, we only show 2× 104 events.
hence the transition regions get wider. Therefore, the shift in a increases dramatically,
compared to measurements with the standard height of the main magnetic field. Moreover,
the impact of the edge effect further depends on the rotation angle φdet of the proton
detector in the x-y-plane. This also reflects the dependence of the edge effect on the
width of the transition regions, as the rotation angle Eq. (5.68) corresponds to a widening
of the projected area (see also Fig. 3.23) by about 10%.
For comparison, Fig. 5.49 shows the influence of the edge effect on a for the full width
of the neutron beam but the simulated beam profile instead. The comparison to Figs. 5.48c
and 5.48d emphasizes the necessity of a precise knowledge of the neutron beam profile
158 CHAPTER 5. DATA ANALYSIS
Table 5.6: Mean proton drift (∆x,∆y) for several different settings of the aSPECT spectrometer. See the text for details.
Imain U1 U2 U8R U8L U16A U16B U17 ∆x ∆y Measurement run
[A] [V] [V] [V] [V] [kV] [kV] [kV] [mm] [mm]
30 800 820 0 -200 -2 -2 -15 −1.25± 3.75 −1.5 ± 4.5 22_05_08/night
30 800 820 -50 -1000 -2 -2 -15 −5 ± 4 −1.5 ± 4.5 21_05_08/night, 22_05_08/teatime
70 0 0 -50 -1000 -4.2 -0.2 -10 −2.5 ± 2 −3 ± 1.5 21_04_08/night
70 800 820 0 -200 -2 -2 -15 −0.75± 1.25 0 ± 1.5 15_05_08/night
70 800 820 0 -200 -3.7 -4.3 -15 −0.5 ± 2.5 0.5 ± 2 16_05_08/night
70 800 820 -50 -1000 -2 -2 -15 −2 ± 2 0 ± 2 19_05_08/night_2, 20_05_08/lunch,
20_05_08/night, 21_05_08/morning
70 800 1000 0 -10 -2 -2 -15 0 ± 2 0 ± 1.5 19_05_08/lExB
70 800 1000 0 -100 -4.2 -0.2 -10 −0.25± 2.75 −3 ± 2 23_04_08/night
70 800 1000 0 -200 -4.2 -0.2 -10 −0.2 ± 2.75 −3 ± 3 12_05_08/night
70 800 1000 0 -2000 -4.2 -0.2 -10 −3.5 ± 2.5 −3 ± 3 24_04_08/night
70 800 1000 -50 -1000 -2 -2 -9 −2 ± 2 0 ± 2 16_04_08/night
70 800 1000 -50 -1000 -2 -2 -10 −2 ± 2 0 ± 2 20_04_08/night, 14_05_08/night
70 800 1000 -50 -1000 -2 -2 -12 −2 ± 2 0 ± 2 27_04_08/night, 28_04_08/night
70 800 1000 -50 -1000 -2 -2 -15 −2 ± 2 0 ± 2 17_05_08/night2, 18_05_08/night,
19_05_08/night_1
70 800 1000 -50 -1000 -3.7 -4.3 -15 −2 ± 2 0.25± 2.25 17_05_08/night
70 800 1000 -50 -1000 -4.2 -0.2 -10 −2.25± 2.25 −3 ± 3 17_04_08/night, 18_04_08/night,
19_04_08/night, 22_04_08/night,
25_04_08/night2, 06_05_08/night,
11_05_08/night2, 13_05_08/night2
70 800 1000 -50 -1000 -4.2 -0.2 -15 −2.25± 2.25 −3 ± 3 18_05_08/morning
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 159
(a) 5mm wide aperture (b) 20mm wide aperture
(c) B0 = 2.177T (d) B0 = 0.933T
Figure 5.48: Relative change of the angular correlation coefficient a for different positions and rota-
tion angles of the proton detector, (a) for the 5mm wide aperture (17_04_08/night),
(b) for the 20mm wide aperture (14_05_08/night), (c) for the full width of the neu-
tron beam (20_05_08/night), and (d) for the full width of the neutron beam but a
reduced height of the main magnetic field (21_05_08/night). With increasing width
of the neutron beam the impact of the edge effect on a decreases, whereas with de-
creasing height of the main magnetic field the shift in a increases dramatically. See
the text and Table 5.6 for details. The error bars show statistical errors only and
are strongly correlated, as each point stems from the same MC simulation (number
of generated events = 108).
in the DV. Hence, we strongly recommend improved measurements of the neutron beam
profile(s) directly in the DV and subsequent analysis of the measured beam profiles by
means of the image plate scanner (cf. also Fn. 5 in Chap. 4), in a further beam time.
Dependence of a on the Location on the Proton Detector
Further measurements at a low-energy, low-flux proton accelerator for detector tests
(PAFF) [237, 238] revealed that the pulse height of the protons and hence the position of
the proton peak changes over the detector [52]. Figure 5.50 shows that the average pulse
height of the protons is reduced close to the edges of the detector. At present, it is still
unclear whether this effect is restricted to a small area around the point investigated with
160 CHAPTER 5. DATA ANALYSIS
(a) B0 = 2.177T (b) B0 = 0.933T
Figure 5.49: Uncertainty in the angular correlation coefficient a due to the knowledge of the neu-
tron beam profile. (a) and (b) show the relative change of a for different positions
and rotation angles of the proton detector. Compared to Figs. 5.48c and 5.48d, these
figures show the influence of the edge effect on a for the simulated instead of the
measured beam profile. See the text and Figs. 5.44 and 5.48 for details. The error
bars show statistical errors only and are strongly correlated, as each point stems from
the same MC simulation (number of generated events = 108).
Figure 5.50: Change of the position of the proton peak over the detector. Figure taken from
Ref. [52].
PAFF. For details the reader is referred to Ref. [52].
With regard to the dependence of the neutrino-electron correlation coefficient a on
the lower integration limit (discussed in Sec. 5.4.1), the spatial distribution of the edge
effect over the detector was investigated. For this purpose, the detector was divided into
twenty 0.5mm wide stripes parallel to the neutron beam. Then, the influence of the edge
effect on a was calculated separately for each stripe. Figure 5.51 shows the results of our
2D MC simulations for measurements with a reduced widths of the neutron beam as well
as the standard settings of our spectrometer. Again, the impact of the edge effect on a
strongly depends on the position of a single stripe. In the case of the 5mm wide aperture,
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 161
(a) Additional aperture (b) Full width
Figure 5.51: Relative change of the angular correlation coefficient a for different locations on the
proton detector, (a) for the 5mm and the 20mm wide aperture and (b) for the full
width of the neutron beam. The figures show the spatial distribution of the edge
effect over the width of the proton detector. See the text and Figs. 5.48 and 5.49 for
details. Number of generated events = 4 × 108 for the 5mm wide aperture, 8 × 108
for the 20mm wide aperture, 16 × 108 for the full width of the neutron beam, and
32 × 108 for the full width of the neutron beam but the simulated instead of the
measured beam profile. Error bars show statistical errors only.
the shift in a reaches in excess of ∆a/a ≈ 700%, compared to ∆a/a = −6.86(39)% over
the entire detector (from Fig. 5.48a). However, for the full width of the neutron beam,
the dependence of a on the location on the proton detector (O (1%)) cannot explain the
observed shift Eq. (5.19) in the order of −10%, cf. Fig. 5.51b.
We note that Fig. 5.51b additionally emphasizes the necessity of a more uniform neu-
tron beam profile. For the rather flat, simulated neutron beam profile (Fig. 5.44a com-
pared to 5.44c), the shift in the angular correlation coefficient a is almost constant over
the proton detector.
Correction for the Edge Effect
As explained in the previous sections, the impact of the edge effect on the neutrino-electron
correlation coefficient a is derived from combined MC simulations, where
• MC simulations in (2D) axially symmetric fields serve to determine the dependence
of a on the position of the proton detector, cf. Fig. 5.48, and
• MC simulations in (3D) non-axially symmetric electric fields serve to determine the
mean proton drift in dependence of the settings of the spectrometer, cf. Fig. 5.47
and Table 5.6.
Table 5.7 presents the results of our combined MC simulations in comparison with the
uncorrected values, aexp, for the angular correlation coefficient a. Obviously, the observed
shifts, ∆aexp/aexp, in the angular correlation coefficient a are in poor or no agreement
with the predicted values, ∆a/a, from our MC simulations. In particular,
162 CHAPTER 5. DATA ANALYSIS
Table 5.7: Relative change of the angular correlation coefficient a for different settings of the aSPECT spectrometer, see Table 5.6 for details. The
third column gives the uncorrected values aexp for a. The fourth and fifth column list the observed changes of aexp in comparison with the
expected values from our MC simulations of the edge effect. See the text for details.
Measurement run ∆y aexp ∆aexp/aexp ∆a/a, simulated Test Comment
[mm] [%] [%]
21_05_08/night 0 -0.0697(38) B0 = 0.933T
20_05_08/lunch, 20_05_08/night 0 -0.1025(28) 31.97± 4.16 −22.82± 0.91+0.35−0.00 vs. B0 = 2.177T
15_05_08/night 0 -0.1092(31) 20mm aperture different
20_05_08/lunch, 20_05_08/night 0 -0.1025(28) −6.58± 4.22 −3.67± 0.74+1.37−0.09 vs. full width lower E×B
18_05_08/morning -3 -0.0978(37) full width,
17_05_08/night2, 18_05_08/night 0 -0.0978(29) 0.00± 4.80 −0.06± 0.81+0.52−0.00 upper E×B
14_05_08/night 0 -0.1118(46) 20mm aperture different U17;
27_04_08/night, 28_04_08/night 0 -0.1050(33) −6.53± 5.49 −3.67± 0.74+1.37−0.09 vs. full width cf. also Fn. 7
13_05_08/night2 -3 -0.1003(43) 20mm aperture,
14_05_08/night 0 -0.1118(46) 10.29± 5.33 1.20± 0.59+0.37−1.41 upper E×B cf. also Fn. 7
13_05_08/night2 -3 -0.1003(43) 20mm aperture
22_04_08/night, 25_04_08/night -3 -0.1014(22) 1.06± 4.74 −2.36± 0.73+0.09−0.61 vs. full width cf. also Fn. 7
17_04_08/night -3 -0.1433(64) 5mm aperture
18_04_08/night, 19_04_08/night -3 -0.1084(28) −32.23± 6.83 −7.58± 0.71+1.07−1.38 vs. full width cf. also Fn. 7
16_04_08/night 0 -0.1358(48) 5mm aperture, different U17;
17_04_08/night -3 -0.1433(64) 5.23± 5.40 1.51± 0.48+3.37−1.30 upper E×B cf. also Fn. 7
16_04_08/night 0 -0.1358(48) 5mm aperture different U17;
20_04_08/night 0 -0.1014(40) −33.93± 7.09 −6.02± 0.72+4.86−2.22 vs. full width cf. also Fn. 7
5.5. INVESTIGATIONS OF SYSTEMATIC EFFECTS 163
• in the case of measurements with a reduced height of the main magnetic field, by a
factor of 3/7, the uncorrected value for a is shifted to more positive values while the
simulations predict a shift to more negative values, and
• in the case of measurements with an additional 5mm wide aperture, i.e., a neutron
beam profile reduced in width by about a factor of 3, the uncorrected value for a is
shifted to much more negative values than predicted by the simulations.
As stated in the previous Sec. 5.4, our data analysis revealed three major problems in
the aSPECT spectrometer:
• a saturation effect of the detector electronics shortly after high-energy electrons
(for details see Sec. 5.4.1). For measurements with a reduced height of the main
magnetic field, this effect is reduced, because of an increased spatial separation of
decay protons and electrons on the proton detector (cf. Sec. 5.5.5).
• a possible charging of the collimation system (see Secs. 5.4.4 and also 5.5.2). As
already mentioned, for a strongly reduced width of the neutron beam, this effect is
suppressed.
• possible Penning discharges in the bottom of the spectrometer (discussed in
Sec. 5.4.3). For the strongly reduced width of the neutron beam, the total neutron
flux decreases by about a factor of 4.5 (cf. Fn. 14). Then also the full electron decay
rate is reduced by about a factor of 4.5 and hence the effect of Penning discharges
in the bottom of the spectrometer is reduced.
and in the case of a reduced height of the main magnetic field, additionally,
• an enhanced violation of the condition for adiabatic transport (discussed in
Sec. 5.5.5).
Hence, the measurements with a strongly reduced width of the neutron beam are domi-
nated by the saturation of the preamplifier. Thus, it is possible to estimate the impact of
the saturation effect on the angular correlation coefficient a from the observed shift in a
in comparison to the predicted value from our MC simulations. This comparison results
in a shift of the neutrino-electron correlation coefficient a due to the saturation of the
preamplifier of
∆a/a = (−26.27± 4.95+1.31−2.49)% (5.72)
in agreement with our correction for the saturation effect Eq. (5.32). Here, the relative
error of 19% is dominated by counting statistics. Further 3D MC simulations, considering
the saturation effect, confirmed a shift in the angular correlation coefficient a due to the
saturation of the preamplifier of
∆a/a = −28.92(31)%. (5.73)
We note that the opposite sign of the shift in a, in the case of measurements with a reduced
height of the main magnetic field, was already discussed in Sec. 5.5.5 (see also Secs. 5.4.3
and 5.4.4).
Finally, to correct for the edge effect, we have to subtract the respective shift in a due
to the edge effect, shown in Fig. 5.48, from the uncorrected values, aexp, for a. For our
164 CHAPTER 5. DATA ANALYSIS
Table 5.8: The uncorrected values, aexp, for a for measurements with the full width of the neutron
beam.
Measurement run rB U2 U8R U8L U16A U16B aexp
[V] [V] [V] [kV] [kV]
16_05_08/night 0.203 820 0 -200 -3.7 -4.3 -0.0953(27)
17_05_08/night2 0.203 1000 -50 -1000 -2 -2 -0.0960(48)
18_05_08/morning 0.203 1000 -50 -1000 -4.2 -0.2 -0.0978(37)
18_05_08/night 0.203 1000 -50 -1000 -2 -2 -0.0996(32)
19_05_08/lExB 0.203 1000 0 -10 -2 -2 -0.0921(51)a
19_05_08/night 0.203 1000/820 -50 -1000 -2 -2 -0.0991(34)
20_05_08/lunch 0.203 820 -50 -1000 -2 -2 -0.1020(45)
20_05_08/night 0.2049 820 -50 -1000 -2 -2 -0.1029(34)
aThe uncorrected value aexp for a was determined by the “maximum−baseline method” instead of from
the fitted pulse height spectra (for details see Sec. 5.1). On average, the data fitting shifts such results to
more negative values, by ∆aexp/aexp = (−2.01± 2.00)%.
standard settings (cf. Table 3.1), the edge effect yields a shift in the neutrino-electron
correlation coefficient a of
∆a/a = (−1.474± 0.145+0.100−0.490)% (5.74)
cf. Fig. 5.48c. Here, the first error stems from the simulation statistics (number of
generated events = 16 × 108) and the latter one from the uncertainty in the position of
the proton detector relative to the rest of the electrode system and the magnetic field.
As a note on the performance of our MC simulations: the simulation of only 1 × 108
events yields to a shift in a of, e.g., ∆a/a = −0.79(58)%, in moderate agreement with
Eq. (5.74). Hence, for further investigations of the edge effect, we strongly recommend to
increase the simulation statistics (by means of MC simulations on a computer cluster). In
addition, we suggest to repeat all investigations concerning the edge effect, in a further
beam time.
5.6 The Antineutrino-Electron Correlation Coefficient a
5.6.1 Analysis
As explained at the beginning of this chapter, the data analysis was performed blinded.
After the data analysis was completed, the data was unblinded. Table 5.8 presents the
uncorrected values, aexp, for a for measurements with the full width of the neutron beam
and from a time when the count rates were not fluctuating non-statistically (cf. Fn. 7). The
values for the neutrino-electron correlation coefficient a given in Table 5.8 are corrected
only for the blind analysis Eqs. (5.1) and (5.2), the dead time of the electronics Eq. (5.10),
and the AP voltage dependent background (for details see Sec. 5.3.1). Those corrections
for systematic effects determined in this thesis are not yet included. Therefore, the quoted
uncertainties on a represent only the statistical errors.
5.6. THE ANTINEUTRINO-ELECTRON CORRELATION COEFFICIENT A 165
The scatter between the individual measurement runs was investigated earlier in
Sec. 5.4.3. In contrast to Sec. 5.4.3, the uncorrected value aexp for a from the mea-
surement run 16_05_08/night is shifted to more positive values. We believe that this is
the result of the reduced drift potential (cf. Sec. 5.3.2), from U8R|U8L = −50V|−1000V to
0V|−200V. But we should not jump to conclusions, as also the drift potential U16A|U16B
and the mirror potential U2 were changed. Indeed, the uncorrected value aexp for a from
the measurement run 19_05_08/lExB, for which the drift potential was drastically re-
duced to U8R|U8L = 0V|−10V, is shifted to even more positive values.
5.6.2 Corrections
In Table 5.9 we list all systematic effects on the neutrino electron-correlation coefficient
a, together with the respective corrections and uncertainties. In addition, we give a
preliminary value for the total uncertainty on a.
Table 5.9: Corrections and uncertainties on the antineutrino-electron angular correlation coeffi-
cient a. See the text for details.
Effect on the angular Relative Relative Relative
correlation coefficient a correction error error Reference
(MC)
[%] [%] [%]
Theoretical corrections
Coulomb correctiona −6.86
Higher-order Coulomba −0.26
Recoil correctiona 0.09 0.07 see also [74]
Radiative correctiona ≈ 2.98 [74] and (2.43)
Weak magnetism +0.02−0.01 from [74]
Fit of MC datab < 0.05
Transmission function
B field gradient −0.001 0.05 (3.46)
B field ratio rB 0.26 0.1 (4.6) (see also [53])
Barrier voltage UA −0.09 0.11 (4.7)
Non-adiabatic motionc 0.014 +0.013−0.026 (5.67) (see also [47])
Mirror potentiald −0.55 +3.48−3.25 (5.42)
Charging collimation 23.23 +22.23−16.33 (5.50)
Background
Dependence on U eA ≈ −1 0.2 from [34]
Background peak 1 −1.14 0.3 [34] and (5.15)
aTheoretical correction already included in the fit function.
bCorrection already taken into account in the data analysis.
cDerived from Table 4.1 by numerical interpolation, with ∆a/a = 75.52 · exp(−3.952T−1 ·B0).
dCorrection only applicable for U2 = 1000V compared to U2 = 820V.
eCorrection already included in the raw data analysis.
166 CHAPTER 5. DATA ANALYSIS
Effect on the angular Relative Relative Relative
correlation coefficient a correction error error Reference
(MC)
[%] [%] [%]
Background peak 2 2.07 0.31 [34] and (5.16)
Penning Dischargesd ≈ 9.29 +5.04−4.61 (5.43)
Detector efficiency
Proton backscattering < 0.22 0.16 (3.56) (see also [52])
Electronic noise 0.035 0.05 (5.12)
Dead time < 4e 0.145 [34, 52], (5.12), (5.14)
Lower integration limite 0.13 0.09 [52] and (5.19)
Trigger efficiency ≈ 0.2 Ref. [34] and (5.20)
Preamplifier saturationf −16.81 +6.00−5.05 (5.32)
Electron backscattering −0.38 +3.56−3.32 (5.40)
Edge effectg −1.45 +0.18−0.49 (5.74)
WF inhomogeneity
in the AP 0.37 0.07 (6.9)
in the DV 0.24 0.05 (6.10)
Proton reflections 0.03 +0.03−0.32 0.08 (6.27)
Absolute WF values +1.01−0.99 0.13 (6.9),(6.10),(6.27),(6.28)
Systematics, included ≈ −0.93 0.09
Systematics, excluded ≈ 5.93 +23.38−17.43 0.28 please note Fn. f
Systematics, excludedd ≈ 14.68 +24.17−18.32 0.28 please note Fn. f
Statistics 1.4 from [34]
5.6.3 Result
As quoted earlier on page 127, we could only set upper limits on the correction of the
problem in the detector electronics, which are too high to determine a meaningful result
from our latest beam time at the ILL. Therefore, our collaboration decided to stop the
data analysis without giving a final or even preliminary value for the antineutrino-electron
angular correlation coefficient a.
From the measurement runs 20_05_08/lunch and 20_05_08/night, e.g., we could
extract a preliminary value for the angular correlation coefficient a of
aprel = −0.1085± 0.0028(stat)+0.0179−0.0240(sys), (5.75)
similar to Stratowa et al. and Byrne et al., but less accurate. However, I merely mention
it as a personal example.
fCorrection only applicable for a lower integration limit of 80 ADC channels.
gCorrection only applicable for the full width of the neutron beam, Imain = 70A, and U16 = −2 kV.
Chapter 6
Investigations of the Patch Effect
In our measurement, the potential difference between decay volume (DV) and analyzing
plane (AP), UA − U0, has to be known precisely to determine the transmission function,
Eq. (3.30). In order to keep systematic uncertainties in the neutrino-electron correlation
coefficient a well below the design accuracy of aSPECT, which is ∆a/a = 0.3%, we
have to know the potential barrier UA (U0 ≡ 0) with an accuracy of better than 10mV,
cf. Sec. 3.1.3. The voltage applied to the AP electrode e14 is monitored by a precise
multimeter, currently an Agilent 3458A. The accuracy of the voltage settings is better
than 5mV, limited by the calibration of the multimeter. However, a variation of the work
function within an electrode or between different electrodes would render the potential
difference UA − U0 uncertain despite the multimeter measurement. Indeed, a variation of
up to 250meV over a distance of several cm was found in a cylindrical sample electrode at
atmospheric pressure and room temperature1, shown in Fig. 6.1. Assuming the potential
barrier UA were measured incorrectly by this amount, the extracted value for a would
be wrong by 3.2(1)%, cf. Sec. 3.1.3. Therefore, surface coatings and/or treatments for
the inner surfaces of our electrode system, which do not show this spatial variation, have
to be found. We suspect that the impurity of our gold surface caused the problem2.
Sputter coated gold surfaces or different surface materials are reported to give better
results [205, 239–241].
To investigate this effect, further measurements with a Kelvin Probe are ongoing.
These measurements are conducted at the University of Virginia in Charlottesville, USA.
After a short introduction to the Kelvin technique, we will present the current status of our
study. Based on the results of our investigations with the Kelvin Probe, we will determine
the impact of the patch effect on the angular correlation coefficient a. More details on the
measurements with the Kelvin Probe can be found in the theses of R. Hodges [242] and
S. McGovern [243].
1This investigation was performed in cooperation with Prof. I. Baikie from KP Technologies Inc. in
Wick, Scotland, using a custom-designed Kelvin probe similar to the one discussed in Ref. [239]. Details
of the specific Kelvin probe and details about the investigation will be published soon.
2E.g., traces of Ca, Ni, Al, Si, Ti, Mg, Na, C, and O were found using a scanning electron micro-
scope (SEM) together with an energy dispersive X-ray (EDX) analysis. The measurement was taken in
cooperation with Dr. J. Huth from the Max Planck Institute for Chemistry in Mainz, Germany.
167
168 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.1: Cylindrical sample electrode. The electrode was made of OFHC copper and was elec-
trolytically coated with 2µm of silver and then 1µm of gold. (a) Morphological im-
age. For circumference measurements, the electrode was held by an aluminum sample
holder which was then mounted onto a rotational platform. Photograph courtesy of R.
Hodges [242]. (b) Kelvin Probe scan1. Work function (WF) topography with 10meV
contour graduations: avg WF 4991meV, RMS WF 44.5meV, PTP WF 178.9meV. R.
Hodges measured the electrode at University of Virginia, after it had been measured
by Prof. I. Baikie, and found a PTP WF value of about 250meV [242].
6.1 Introduction
The work function (WF) is the minimal energy required to extract one electron from the
surface of a conducting material to a point outside the metal with zero kinetic energy. The
WF is continuous across the interior of the material, however at the surface the electron
energy is influenced by the exact state of the surface, e.g., type orientation and direction
of the outer atoms and molecules. Thus different crystallographic orientations of the same
material may have different WF, see Table 6.1. As the electron has to move through the
surface region, its energy is influenced by the optical, electric, and mechanical characteris-
tics of this region. Hence, the WF is modified by absorbed or evaporated layers [245, 246],
surface reconstruction, surface charging, oxide layer imperfections, surface and bulk con-
tamination, etc. The WF may vary over the surface of a polycrystalline material due to
regions of different crystal orientation and over the surface of an alloy or compound due
to non-uniform segregation of the elements [247]. Such differences in WF over regions of
a surface cause the so-called patch effect.
6.1.1 Surface Analysis
The WF is an extremely sensitive indicator of surface condition and can therefore be em-
ployed in many detection-based scenarios. Examples include WF/topography variations
which occur in, e.g., adsorption, biotechnology, charge analysis, coatings, corrosion, nan-
otechnology, semiconductors, solar cells, surface contamination, surface chemistry, and
surface photo-voltaic. Scanning probe microscopy (SPM) covers several related technolo-
gies for imaging and measuring surfaces on the nanometer scale. SPM technologies share
6.1. INTRODUCTION 169
Table 6.1: Electron work function (WF) of selected elements from the CRC Handbook of Chem-
istry and Physics [244]. As can be seen in the second column, the WF usually differs for
each face of a mono-crystalline sample. Since the WF is dependent on the cleanliness of
the surface, measurements reported in the literature often cover a considerable range.
The third column contains selected values for the WF which may be regarded as typical
values for a reasonably clean surface. Values in parentheses are only approximate. In
the fourth column, the method of measurement is indicated for each value. The follow-
ing abbreviations are used: polycr (polycrystalline sample), PE (photoelectric effect),
and CPD (contact potential difference).
Element Plane WF [meV] Method Element Plane WF [meV] Method
Ag 100 4640 PE Fe 100 4670 PE
110 4520 PE 111 4810 PE
111 4740 PE Mg polycr 3660 PE
Al 100 4200 PE Na polycr 2360 PE
110 4060 PE Ni 100 5220 PE
111 4260 PE 110 5040 PE
Au 100 5470 PE 111 5350 PE
110 5370 PE Pt polycr 5640 PE
111 5310 PE 100 5840 PE
Be polycr 4980 PE 111 5930 PE
C polycr (5000) CPD 320 5220 PE
Ca polycr 2870 PE 331 5120 PE
Cr polycr 4500 PE Rh polycr 4980 PE
Cu 100 5100 PE Si n 4850 CPD
110 4480 PE p 100 (4910) CPD
111 4940 PE p 111 4600 PE
112 4530 PE Ti polycr 4330 PE
the concept of scanning an extremely sharp tip (3-50 nm radius of curvature) across the
object surface. The most commonly used SPM techniques are:
Atomic Force Microscopy (AFM): measures the interaction force between the tip
and surface.
Scanning Tunneling Microscopy (STM): measures a weak electrical current flowing
between tip and sample as they are held a very distance apart.
In the present thesis, a Kelvin Probe (KP) was used to measure the WF of primarily
gold- and platinum-plated samples. Although not as well known as other surface analysis
techniques, the KP has undergone a dramatic renaissance over the last few years. The KP
is a non-invasive technique, yet it is extremely sensitive to changes in the top-most atomic
layers, such as those caused by deposition, absorption, corrosion, and atomic displacement.
170 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.2: (a) Main components of the computer (PC) controlled ambient Scanning Kelvin Probe
system SKP5050 [250]. The PC houses the digital oscillator (which powers the voice
coil actuator), data acquisition system, and motorized (x, y, z) stage controller. The
signal is derived from a low-noise, high-gain current to voltage (I/V) converter mounted
close to the tip. (b) Gold probe tip measuring a reflective aluminum sample. Photo-
graph courtesy of S. McGovern [243].
6.2 Operation of the Kelvin Probe
The Kelvin Probe is a non-contact, non-destructive vibrating capacitor device which mea-
sures the WF difference between a conducting specimen and a vibrating probe tip, as
shown in Fig. 6.2b. The KP measures the average WF of the sample under the probe tip
without the bias toward low WF patches characteristic of photo and field emission. The
particular KP at University of Virginia used for these measurements is a custom-designed
ambient Scanning Kelvin Probe (SKP) system [242, 243], similar to the one shown in
Fig. 6.2a. The probe tip used is a gold disk electrode of diameter 2mm, shown in Fig. 6.2b,
which is scanned above the sample surface using a computer controlled (x, y, z) scanning
system. This allows the measurement of the WF, with a resolution of (1− 3)meV [248]3,
to be spatially resolved across the surface. A general description of the probe operation
can be found in Ref. [249] and more details in its operating manual [248]. Here, we will
only give a brief description.
6.2.1 Theory
The series of schematic diagrams on the following page illustrate the principles of local
WF measurement using a KP. In Figure 6.3a, the KP tip has a Fermi level, ETip, which
differs from that of the sample, ES. The difference in Fermi levels also implies a difference
in the WF, Φ. If an external electrical contact is made between probe tip and sample, the
3In the case of a 50µm tip, the WF resolution is (5− 10)meV only.
6.2. OPERATION OF THE KELVIN PROBE 171
(a) (b)
(c)
Figure 6.3: Electron energy level diagrams of two conducting specimens, Tip and Sample (S), (a)
without contact, (b) with external electrical contact, and (c) with inclusion of the
backing potential Vb. ΦTip and ΦS are the work functions of the materials, and ETip
and ES represent their Fermi levels. In (b) the surface charge is related to the contact
potential difference, VCPD, through Q = VCPDCKP, where CKP is the Kelvin probe
capacitance. If tip and sample are connected by the external emf, Vb, and vibrated,
then the current IKP = dQ/dt = (VCPD + Vb) dCKP/dt flows.
Fermi levels equalize, resulting in the generation of an electrical charge on the respective
surfaces, as shown in Fig. 6.3b. This surface charge gives rise to a potential difference,
termed the contact potential difference (CPD), VCPD, which relates to the difference in
WF, ∆Φ = ΦS − ΦTip, such that:
1
VCPD = ∆Φ, (6.1)
e
If a variable backing potential, Vb, is included in the external circuit, as shown in Fig. 6.3c,
then the surface charge will become zero at the unique point where:
Vb = −VCPD. (6.2)
The KP deduces the point of zero charge by vibrating the probe tip in a vertical plane
perpendicular to the sample surface and measuring the AC current flow, IKP, that results
if a surface charge, Q, exists:
d
IKP(t) = Q(t). (6.3)
dt
172 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Figure 6.4: Typical Kelvin Probe output signal. As the probe oscillates above the sample the
voltage change is recorded. The periodic signal Eq. (6.8) is shown with the modulation
index set to  = 0.7.
The surface charge is related to the CPD and the external emf, Vb, through
Q(t) = (VCPD + Vb)CKP(t), (6.4)
where CKP is the KP capacitance. If probe tip and sample are assumed to have parallel-
plate geometry, then the KP capacitance is given by
A
CKP(t) = 0 , (6.5)
d(t)
where 0 is the dielectric constant, A is the area of the capacitor, and d is the separa-
tion distance of the plates. As the probe tip is oscillated above the sample surface, the
separation distance can be written as
d(t) = d0 + d1 sin (ωt) , (6.6)
where d0 is the distance at rest between probe tip and sample, d1 is the amplitude of
oscillation, and ω is the angular frequency of vibration. By substituting  for d1/d0,
termed the modulation index, and C0 for 0A/d0 in Eq. (6.5) we get
C0
CKP(t) = . (6.7)1 +  sin (ωt)
We successively substitute the result into Eqs. (6.4) and (6.3) to get the voltage change,
given by the Ohm’s law, Vout(t) = IKP(t)R:
cos (ωt)
Vout(t) = − (VCPD + Vb)RC0ω
(1 +  sin (ωt))2
sin (ωt+ θ)
= (VCPD + Vb)RC0ω , (6.8)
(1 +  sin (ωt))2
where R is the I/V converter feedback resistance and θ is the phase angle. A typical KP
output signal is shown in Fig. 6.4. When the peak-to-peak voltage, VPTP, is equal to zero,
then the WF difference is simply equal and opposite to Vb, see also Eq. (6.1).
6.2. OPERATION OF THE KELVIN PROBE 173
Figure 6.5: Peak-to-peak voltage VPTP versus backing potential Vb. As can be seen from Eq. (6.8)
this is a straight line. The slope is termed the Gradient and the intersection with the
Vb-axis defines the measured work function value.
6.2.2 The “off-null” Technique
Traditional phase sensitive methods utilize a lock-in amplifier to detect the null output
condition. However, these methods have the disadvantage that at the balanced point, the
signal-to-noise ratio (SNR) reaches a minimum and the noise creates an offset voltage. A
better method is to set the backing potential Vb to a range of voltages around the balanced
point and then extrapolate to the null signal condition, as shown in Fig. 6.5, thus working
to a high SNR. In this way, changes in VCPD can be determined to sub-mV resolution.
This “off-null” detection method was invented by Prof. I. Baikie [251].
6.2.3 Work Function Measurement
Since KP measurements can only detect the CPD, actual WF measurements are only
possible through calibration. In other words, the KP needs to be calibrated against a
surface with known WF. The challenge in this respect is that under ambient conditions
it is hard to generate a surface with a defined WF. Hence KP measurements are more
reliable in vacuum, where a well-defined surface can more reproducibly be generated and
controlled. In aSPECT, we only have to know precisely the potential difference between
DV and AP, UA − U0, to determine the transmission function, Eq. (3.30). Therefore, the
calibration against a surface with known WF can be trade off for a SKP which moves
between DV and AP and measures the WF difference between the respective electrode
surfaces and the vibrating probe tip.
Within this thesis, an ambient SKP was used to measure the WF difference between
the samples and the probe tip. The WF measurements were performed at atmospheric
pressure and room temperature. Preliminary calibration and experimentation of the SKP
followed the instructions given in the operating manual [248]. WF measurements are
prone to difficulties caused by stray charges, parasitic capacitance effects (discussed in
Ref. [249]), and external mechanical vibrations. Stray charges were reduced by grounding
the system to a common potential, except for the samples and the probe tip. External
mechanical vibrations were reduced by weighing down the SKP stage with lead bricks.
174 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
6.2.4 Reference Sample
In Ref. [252] an error deduction method was suggested to eliminate the influence of space
charge on the performance of the KP by introducing a reference sample. As mentioned
above, this method is also valid to cancel the error due to changes in the WF of the probe.
The particular KP used for these measurements was delivered with a reference sample
with associated SKP topography. It is a polished aluminum substrate, half of the surface
is gold-plated. The reference sample was characterized in Ref. [242]. A WF difference
of 1404meV to the theoretical difference of about 1400meV was found4. In Ref. [243]
the performance of a reference sample was tested5. A linear slide was introduced which
moves back and forth between sample and reference sample. The probe tip remained at
the same place and measurements, of one point on each surface, were taken for several
hours. Unexpectedly, it has been found that the difference in WF values was not constant
in time. We suspect that the surfaces are effected in different ways over time. Primarily,
the target of our investigation is WF variation. In this regard, comparison of the standard
deviation (RMS) on the WF for different samples is independent of the average WF value
of a single sample. Consequently, further use of a reference sample was eliminated.
6.3 Samples, Data, and Results
The present thesis focused on the investigation of primarily gold and platinum coated
oxygen-free high thermal conductivity (OFHC) copper samples, although others have been
characterized. Table B.1 in App. B presents a summary of the samples studied to date
including details of the substrates, surface treatments, and coatings6. The deposition
method mainly used was electrolytically coating, although samples were fabricated by
electro-chemical coating and physical vapor deposition (PVD), i.e., namely sputtering.
Altogether, 58 samples were manufactured for measurement with the SKP, from what 17
samples remain to be scanned7. Another 18 samples are in preparation for sputtering.
The majority of the samples were coated off-site and then shipped to the University of
Virginia. Except for the glass substrate, the samples were 50 × 50mm2 by 1mm in size.
This size was chosen since it fitted with the SKP (x, y) translation of 50× 50mm2 in the
standard version. The samples were studied individually.
6.3.1 Details of the Samples
The samples chosen for study reflect the various possible materials for the electrodes,
their surface treatments, and coatings to be used in aSPECT. The previous electrodes
were made of OFHC copper and electrolytically coated with 2µm of silver8 and then 1µm
of gold. In order to keep systematic uncertainties in the angular correlation coefficient
a well below 0.3%, additionally, the residual gas pressure has to be ≤ 10−8 mbar and
we have to know the size of the magnetic field B0 in the DV (≈ 2.2T) and BA in the
AP (≈ 0.44T) with a relative accuracy of 10−4 (for details see Sec. 3.1.3 and Fig. 3.5).
Suitable candidates for the substrate material are therefore ultra-high vacuum (UHV)
4The polishing process increases the rate of oxidation which decreases the WF of aluminum from
4200meV (cf. Tab. 6.1) to about 4000meV [205, 253].
5In Ref. [243] sample Cu25 (cf. Sec. 6.3) was used as a reference sample for its twin Cu26 instead.
6Details of the samples coated in the USA can be found in Refs. [242, 243].
7Unfortunately, the KP scans could not be finished within the Master’s thesis of S. McGovern.
8The renewed and over-coated electrodes e3 – e6, e12, and e14 were coated with 10µm of silver instead.
6.3. SAMPLES, DATA, AND RESULTS 175
Figure 6.6: Work function (WF) Φ versus surface roughness (SR) Rq of copper, theoretical curve
and experimental results. The WF of copper decreases with increasing SR. Figure
taken from Ref. [254]. Different crystallographic orientations are emphasized in red
for comparison only.
compatible, non-magnetic, and resistant to chemical reactions, such as OFHC copper,
titanium, glass, silicon, and sapphire9. Candidates for the surface coating are additionally
conductive, such as gold, platinum, ruthenium, and graphite10. Diffusion of copper in gold
can be prevented by depositing an interlayer between the toplayer and the OFHC copper
substrate. The various intermediate layers were chosen from adhesion, conductivity, and
smoothness considerations. In our application, the standard interlayer, nickel, is excluded
as nickel is ferromagnetic.
In addition, materials must be free of cracks and crevices which can trap cleaning
solvents and become a source of virtual vacuum leaks later on. The (unpolished) cop-
per and titanium substrates have a surface roughness (SR) better then 210 and 400 nm
Ra (arithmetic average roughness value) plus 2.5 and 5.5µm Rmax (maximum roughness
depth), respectively (for details see Table 6.5). With smaller SR, the coating adhesion
but also the wear rate increase. In Ref. [254] it was reported that the WF of copper
decreases with increasing SR, as shown in Fig. 6.6. We note that in Ref. [254] all samples
were annealed to eliminate the effect of deformation from polishing on the WF. In order
to increase the coating adhesion and, simultaneously, avoid an anisotropic WF, polishing
techniques were investigated. Where possible, the substrate surface was polished (mechan-
ically and/or by hand) to a mirror finish, as shown in Fig. 6.7a. Thereby non-magnetic
polishing compounds and buffing wheels have been used only.
The electrodes used previously in aSPECT had a cylindrical shape, as shown in
Fig. 6.1a, with the exception of the DV electrode gr. The pipes used were formed by
rolling OFHC copper plates and welding the seam. To get an idea, whether and if so how
the fabrication alters the metal’s crystal lattice and/or WF, processing techniques were
9p-type 100 silicon wafers coated with 1µm of titanium and then 0.1 and 0.2µm of platinum, respec-
tively, were investigated in Ref. [243]. Sapphire wafers were not yet studied.
10In Ref. [205] graphite coating was used to reduce charging inside the beam tube. Attempts to coat the
inner surfaces of our electrode system with colloidal graphite failed. The graphite did not adhere to the
surface. Even preheating the substrate and/or drying in the oven for up to one hour could not stabilize
the adhesion. In contrast, Electrodag E graphite was successfully tested on thin copper and stainless steel
foil sheets in Ref. [242]. R. Hodges found PTP WF values of less than 10meV for single traces.
176 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
investigated. In a first step, copper plates were chopped and subsequently connected again
(by means of welding or brazing), as shown in Fig. 6.12a. The cold forming may require
additional normalizing with subsequent stress relieve annealing. This could be part of
further investigations.
Most of the samples were doubly manufactured, i.e., twins of the same substrate,
surface treatment, and coating (method, material, and layer thickness) were produced.
Samples Cu1, Cu2, Cu7, Cu8, Cu21, Cu22, Ti1, and Ti2 were manufactured at a time when
the ion source at University of Mainz, normally used to clean the substrate surface before
sputtering, was inoperative. To get an impression, how important the cleaning process
is, only samples Cu1, Cu8, Cu21, and Ti2 were cleaned with isopropyl alcohol before
sputtering. Thus, these sputtered samples are no real twins. In addition, there was one
processing technique which could not be studied. Samples Cu5 and Cu6 cannot be scanned
as a tracking error occurs due to the weld seam. Besides, there is one surface coating which
will not be studied. This is a titanium nitride (TiN) coating on copper. The sputtering
on samples Cu39 to Cu42 failed, especially sample Cu39 fell down during the sputtering
process. The company suspects that either the temperature of the substrate [255] or
remnants of the polishing compound [256] caused the problem. To get an idea, how well
an electrolytically coating adheres to the TiN, sample Cu41 was then nickel-plated. As
expected, the nickel does not adhere to the TiN surface.
6.3.2 Kelvin Probe Data
Two types of data were taken, namely spatial and temporal:
• For temporal measurements, the probe tip remained at the same place and measure-
ments were taken for several hours typically. In earlier runs an enigmatic long-term
drift of about (0.5 − 1.0)meVmin−1 was observed, independent of the sample ma-
terial [242]. In Ref. [243] it has been found that this long-term drift was mainly due
to relative humidity and temperature fluctuations11.
• For spatial scans, a raster of points was taken. The KP moves 50 steps of 0.4mm
per step in y, holding x constant, then moves 1.6mm in x and steps back along
the y-direction for 10 steps12. The whole raster therefore took data of 500 pts and
covered a 14.4mm by 19.6mm rectangular area13. To avoid edge effects, we did
not scan closer than 5mm to the edges of the samples. At each position the CPD
between probe tip and sample was measured once14. It took approximately 1 h to
complete one scan. It is claimed that in the majority of measurements, this trade-off
11The KP in Ref. [242] exhibited large noise caused by vibrations of the wires leading from the probe
tip amplifier to the housing of the drive shaft. This additional noise prevented R. Hodges from observing
the influence of relative humidity and temperature fluctuations on the drift.
12According to the instruction manual [248], one translation step of the stepper motor corresponds to
400 nm. The scan parameter were set to 4000 and 1000 translation steps in x and y, respectively. This
corresponds to the specified steps of 1.6mm and 0.4mm in x- and y-direction, respectively. Later, it was
found that one motor step corresponds to (631 ± 10) nm instead [257]. Consequently, all x- and y-axis
scalings would have to be multiplied by about a factor of 1.6, and therefore, e.g., the scans would have
covered a 22.7× 30.9mm2 area instead.
13Later high-resolution scans took data of 2500 pts and covered a 19.6× 19.6mm2 area. In such cases,
the KP moves 50 steps of 0.4mm per step in x instead. Please see also Fn. 12.
14In Ref. [240] at each position the CPD was measured 11 times and averaged. As of August 2010, the
KP Technologies proprietary software does not allow for averaging at each point of a scan.
6.3. SAMPLES, DATA, AND RESULTS 177
between signal averaging, raster size, and measurement duration was sufficient to
keep the drift to under (3− 5)meV [258].
In Table 6.2 we list all the samples studied with spatial scans. The WF topographies
were all recorded with the same SKP measurement parameters. In particular, the track-
ing was set to keep the gradient at (300 ± 5) a.u. for each measurement point, i.e., the
separation distance in Eq. (6.6) at d0 ≈ 0.5mm, with the exception of samples Cu53 and
Cu54. However, the standard deviation of the gradient values (RMS Grad) reached in
excess of 40 a.u., see also Secs. 6.3.3 and 6.3.6. To reduce the influence of surface charges
on the measured WF value, the samples were prepared for measurement exclusively with
compressed air dusting15. In some cases there are two or even three sets of data of the
same day for one sample. These were taken on the same area of the sample, so that we
averaged over the sets, as indicated in the last column.
Table 6.2: Summary of Kelvin Probe scans. Samples Cu5 and Cu6 cannot be scanned as a tracking
error occurs due to the weld seam. Samples Cu24, Cu35 to Cu38, Cu43 to Cu46, Cu55
to Cu58, and Ti9 to Ti12 remain to be scanned. All other samples are in preparation
as stated in Table B.1. See text and Table B.1 for details. The first column gives the
name of the sample. The second and third column list the gradient (Grad) tracking
data. The fourth to sixth column list the work function (WF) data. And the last
column gives a short comment, if appropriate. The following abbreviations appear:
avg (average value), RMS (standard deviation), and PTP (peak-to-peak value).
Ref.no. avg Grad RMS Grad PTP WF avg WF RMS WF Comment
[a.u.] [a.u.] [meV] [meV] [meV]
Cu25 301.5 22.2 49.0 311.0 6.6 avg of 3 scans
Cu26 301.1 20.4 54.0 280.0 9.0 avg of 2 scans
Cu12 301.0 21.4 54.0 35.5 10.9
Cu11 301.0 21.2 86.0 31.3 14.2
Cu20 301.2 21.0 107.0 136.7 14.3
Cu4 301.3 17.8 64.0 -11.0 14.7
Cu23 301.0 16.3 66.0 82.9 15.0
Cu32 305.0 64.7 80.0 13.5 16.2
Cu34 301.4 25.3 98.0 -186.6 18.0
Cu29 301.5 21.6 100.0 -45.2 18.0
Cu33 306.0 38.2 98.0 13.5 20.7
Cu3 301.4 20.7 129.0 -91.3 21.2
Cu30 301.4 22.4 212.0 -60.5 21.2
Cu13 301.0 18.1 134.0 -21.7 21.7
Cu27 302.0 23.5 106.0 294.7 22.4
Cu9 301.1 20.2 148.0 -192.9 25.7
15In Ref. [243] it has been found that ethyl alcohol wipes are the most effective cleaning technique,
since they removed visible contaminants, left by compressed air, without obvious damage to the sample
surface. In later measurements with alcohol cleaning, the sample was covered and allowed to regain WF
value stability overnight before SKP measurement.
178 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Ref.no. avg Grad RMS Grad PTP WF avg WF RMS WF Comment
[a.u.] [a.u.] [meV] [meV] [meV]
Cu14 300.6 15.8 153.0 -83.1 26.8
Cu18 302.1 24.3 228.0 23.0 33.5
Cu31 301.0 18.2 164.0 -77.2 37.3
Cu15 301.0 17.7 191.0 67.4 39.6
Cu1 ≈ 40 data from [259]
Cu16 301.4 22.8 263.0 257.9 41.0
Cu19 301.0 20.5 196.0 36.8 42.8
Cu7 302.0 24.7 281.3 227.7 46.7 avg of 3 scans
Cu54 259 55 ≈ 233 -4 48 data from [257]
Cu8 301.5 24.0 291.5 6.5 51.5 avg of 2 scans
Cu10 300.8 17.6 206.0 -70.8 53.2
Cu17 302.0 22.8 257.0 110.6 56.8
Cu53 254.7 27.4 ≈ 195 -74.3 58.3 avg 3 scans [257]
Cu21 301.0 23.0 278.0 -75.0 63.0
Cu28 301.9 22.2 248.0 166.3 65.3
Cu2 ≈ 70 data from [259]
Cu22 ≈ 200 data from [243]
Ti1 301.0 20.0 185.5 -56.5 32.0
Ti2 293.0 37.5 202.0 8.0 36.5
Glass1 301.0 16.8 79.0 19.1 8.8 avg of 3 scans
Glass2 301.0 12.5 109.5 103.5 17.1 avg of 2 scans
The present investigation showed that a platinized sample (Cu25) achieved the lowest
standard deviation of the WF values over all the points measured over its surface (RMS
WF)16. The sample, shown in Fig. 6.7a, was made of OFHC copper, hand-polished to
a mirror finish, and electrolytically coated with 10µm of silver and then 0.2µm of plat-
inum17. Although the peak-to-peak spread (PTP WF) is in excess of 10meV, the RMS
WF value is 6.6meV, see also Figs. 6.7b, 6.7c, and 6.8a. The RMS WF value of its twin,
sample Cu26, was only a few meV higher at 9.0meV.
6.3.3 Gradient Tracking Error
In Ref. [243] an algorithmic error in the KP Technology Ltd. proprietary software was
identified: The gradient tracking is deficient.
16Since the scans are stepped in 0.4mm steps in y-direction and the probe size is 2mm, each value
recorded is not uncorrelated with the next, and this complicates the interpretation of the RMS of the
whole data set. To get an idea of the significance of this correlation, in Ref. [240] the RMS has been
calculated for a reduced set of data points. It has been found that these values were not dissimilar to the
RMS of the full data set, and so to the required precision, we can use the RMS of the full data set.
17Please note that in Ref. [243] the thickness of the platinum layer is misstated.
6.3. SAMPLES, DATA, AND RESULTS 179
(a) (b)
(c)
Figure 6.7: Sample Cu25 (best result). The sample was made of OFHC copper, hand-polished
to a mirror finish, and electrolytically coated with 10µm of silver and then 0.2µm of
platinum. (a) Morphological image. (b) Work function (WF) topography with 10meV
contour graduations. Color scheme applied for comparison with the cylindrical sample
electrode shown in Fig. 6.1b. (c) WF Histogram. Different Kelvin Probe scan than
what appears in Table 6.2. Scan with 2500 pts over 5 hours: RMS WF 10.6meV, PTP
WF 71.0meV, cf. Table 6.3 (February 2010).
When the KP moves from one point to the next, in general, the separation distance d0
will change, due to variations in surface height, and the gradient will be altered. Actually
it was thought that the tracking adjusts to hold the gradient at the chosen value, equiva-
lent to holding the probe tip-to-sample distance d0 constant. For spatial scans, the probe
is moved laterally over the sample surface and, instead of first finding the preset gradient,
the measurement is taken at the current gradient. Only after the WF measurement the
tracking of the targeted gradient value is completed. Keeping the step size small reduces
how much the separation distance d0 can change and accordingly how much the gradient
can fluctuate. Indeed, removing the effect by post-processing the data would not be a
straightforward procedure. For smooth surfaces, the influence of the gradient fluctuation
on the measured WF topographies is small and at the level of the signal noise, as can be
seen in Fig. 6.8, along y = 2 and y = 16. With increasing SR the gradient fluctuations in-
crease and may affect the WF topographies. Altogether, the SKP still provides meaningful
data, although the gradient tracking error becomes a limiting factor for precision.
180 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.8: Sample Cu25 (best result), see Fig. 6.7 for details. (a) Work function topography with
5meV contour graduations. The pink contour line is shown to emphasize the temporal
stability of Kelvin Probe (KP) scans over a week, cf. Fig. 6.9. (b) Gradient topography
with non-equidistantly contour graduations. Scan with 2500 pts over 5 hours. The
tracking was set to keep the gradient at (300 ± 5) a.u. for each measurement point.
The comparison with (a) shows how the noise in the KP, e.g., the gradient fluctuation
(tracking error) along y = 2 and y = 16, becomes a limiting factor for precision.
(a) (b) (c)
Figure 6.9: Sample Cu25 (best result), see Fig. 6.7 for details. Temporal stability of the Kelvin
Probe scans over one day. The work function topographies, with 5meV contour grad-
uations, were recorded on (a) January 28, 2010 at 9:50 pm, (b) January 29, 2010
at 0:13 am, and (c) January 29, 2010 at 1:59 pm. Scans with 500 pts over 1 hour.
The pink contour line is shown to additionally highlight the temporal stability over a
week, cf. Fig. 6.8a recorded on February 4, 2010 at 10:38 am. The comparison between
(a)−(c) and Fig. 6.8a shows that the absolute values of the measured WF change, due
to the drift described on page 176 under “temporal measurements”, whereas the WF
topographies are comparable.
6.3. SAMPLES, DATA, AND RESULTS 181
Table 6.3: Temporal stability of Kelvin probe scans over months and over a day. For these scans,
a raster of 500 points was taken, except for the latest measurements of Cu12 and
Cu25. The first column gives the name of the sample. The second column includes
the measurement dates. The third and fourth column list the gradient (Grad) tracking
data. The fifth to eight column list the work function (WF) data. And the last
column gives a short comment, if appropriate. The following abbreviations appear:
avg (average value), RMS (standard deviation), and PTP (peak-to-peak value).
avg RMS PTP avg RMS ∆ avg
Ref.no. Date Grad Grad WF WF WF WF Comment
[a.u.] [a.u.] [meV] [meV] [meV] [meV]
Cu3 Aug 09 301.4 20.7 129.0 -91.3 21.2
Jan 10 -11.4 21.5 80 data from [243]
Jan 10 -12.8 24.0 data from [243]
Cu10 Aug 09 300.8 17.6 206.0 -70.8 53.2
Jan 10 -58.4 36.2 12 data from [243]
Jan 10 -34.5 33.8 data from [243]
Cu11 Aug 09 301.0 21.2 86.0 31.3 14.2
Jan 10 301.1 22.5 66.0 -30.6 11.3 -62
Jan 10 300.7 19.2 51.0 -32.7 9.3
Cu12 Aug 09 301.0 21.4 54.0 35.5 10.9
Jan 10 300.9 25.9 129.0 -51.8 17.5 -87
Jan 10 301.0 26.2 119.0 -65.0 11.7
May 10 301.0 21.0 75.0 -61.0 11.8 4 2500 pts
Cu20 Aug 09 301.2 21.0 107.0 136.7 14.3
Jan 10 160.6 16.0 24 data from [243]
Jan 10 163.6 15.7 data from [243]
Cu25 Jan 10 302.0 22.4 44.0 310.0 7.0
Jan 10 301.6 22.9 46.0 312.9 6.5
Jan 10 301.0 21.4 56.0 210.0 6.3
Feb 10 301.0 19.0 71.0 259.0 10.6 -51 2500 pts
In addition, Fig. 6.8b shows a stripiness in the direction in which the SKP is stepped
to take data, i.e, along the y-direction. The stripy effect is seen in all gradient data
and is caused by mechanical hysteresis. The stepper motor has some backslash, i.e.,
the separation distance d0 changes depending on the direction of motion, and hence the
gradient alternates between higher and lower values. This effect, without present peaks,
was also observed in Ref. [240].
6.3.4 Temporal Stability
In Table 6.3 we list all the samples studied in at least two different months and at least
twice one day. As mentioned earlier in Sec. 6.2.3, these measurement data were taken in
182 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a)
(b) (c)
Figure 6.10: Sample Cu12 (best gold-plated result). The sample was made of OFHC copper,
hand-polished to a mirror finish, and electrolytically coated with 10µm of silver and
then 1µm of gold. Temporal stability of the Kelvin Probe (KP) scans over months.
The work function (WF) topographies, with 5 (10 in (b))meV contour graduations,
were recorded in (a) August 2009, (b) January 2010, and (c) May 2010. The KP
scan (a) was rotated by - 90 ° as we cannot exclude the possibility that the sample
was scanned in a different orientation than (b) and (c). Scans with 500 (2500 in
(c)) pts over 1 (5) hour(s). The comparison between (a)−(c) shows that the absolute
values of the measured WF change, due to the drift described on page 176 under
“temporal measurements”, whereas the WF topographies are comparable. Please
note the different contour scales.
the open laboratory and not in the isolated glove box discussed in Ref. [243]. The average
values of the measured WF (avg WF) change, due to the drift described on page 176
under “temporal measurements”. The RMS WF values, however, remain stable to about
±3.0meV or less, with the exception of sample Cu10. The reasons for the high temporal
stability of the KP scans are on the one hand the reproducibility of the SKP and on the
other hand the storage stability of noble metals under standard laboratory conditions.
Figures 6.9 and 6.10 illustrate the temporal stability of the KP scans over one day and
over eight months, respectively. The WF topographies are comparable.
We note that sample Cu12 (best gold-plated result) was measured by Prof. I. Baikie in
Scotland in Spring 2010, after and before it had been measured at University of Virginia in
6.3. SAMPLES, DATA, AND RESULTS 183
Table 6.4: Reproducibility of those Kelvin probe samples which were produced at least twice.
As input for our comparison we have used the measurement data from Table 6.2. For
details on the samples see App. B. The first column gives the name of the samples. The
second column specifies the respective manufacturer. The third and fourth column list
the work function (WF) data. The fifth and sixth column specify the coating, listed
from substrate surface to outer surface. The last column gives a short comment, if
appropriate. All abbreviations are standard nomenclature for chemical elements. In
addition, the following abbreviation is used: RMS (standard deviation).
avg RMS
Ref.no. Co. RMS WF RMS WF 1st layer 2nd layer Comment
[meV] [meV] [µm] [µm]
Cu3-Cu4 Adler 17.8 4.6 Ag 10 Au 1 polished
mechanically
Cu9-Cu10 Adler 39.5 19.4 Ag 10 Au 2
Cu11-Cu12 Adler 12.6 2.3 Ag 10 Au 1
Cu13-Cu14 Adler 24.3 3.6 Ag 10 Au 5
Cu32-Cu33 Adler 18.5 3.2 Ag 10 Au 1 brazed seam
Cu15-Cu17 Gierich 45.8 9.6 Ag 10 Au 2
White
Cu18-Cu20 Gierich 30.2 14.5 bronze 1 Pt 1
Cu25-Cu26 Sigrist 7.8 1.8 Ag 10 Pt 0.2
Cu27-Cu28 Sigrist 43.9 30.3 Ag 10 Au 2
Cu53-Cu54 Sigrist 53.2 7.3 Ag 10 Pt 0.2
Cu25-Cu26
&Cu53-Cu54 Sigrist 26.6 Ag 10 Pt 0.2
Winter 2009 and Spring 2010, respectively. Despite differences in scan parameters, Prof.
I. Baikie confirmed the results of the SKP measurements in Virginia. In particular, Prof.
I. Baikie found a RMS WF value of 10.8meV [243]. At University of Virginia, an averaged
RMS WF value of 13.0meV was found; where the average over four measurements includes
one measurement of 10.9meV. This shows that the SKP used for our investigation is close
to the sensitivity of the establihed equipment used by the founder of KP Technology Inc.
6.3.5 Reproducibility of the Samples
As mentioned earlier in Sec. 6.3.1, most of the samples were doubly manufactured. In some
cases even triplets or a quadruplet of the same substrate, surface treatment, and coating
were produced. More precisely, the quadruplet (Cu25, Cu26, Cu53, and Cu54) consists of
two pairs of twins manufactured in February 2009 (Cu25 and Cu26) respectively August
2010 (Cu53 and Cu54). In Table 6.4 we list all the samples produced at least twice. The
RMS WF values of seven out of ten twins (triplets) are reproducible to about ±10meV or
less. Unfortunately, there is no obvious explanation why half of the twins (triplets) exceed
±5meV.
184 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.11: Sample Cu28 (worst result). The sample was made of OFHC copper, hand-polished
to a mirror finish, and electrolytically coated with 10µm of silver and then 2µm of
gold. (a) Morphological image. (b) Work function (WF) topography with 15meV
contour graduations. Scan with 500 pts over 1 hour: RMS WF 65.3meV, PTP WF
248.0meV. The sample lacks of the desirable homogeneity in WF, even though it
was polished and coated by the same company as the best sample (Cu25). The large
fluctuations in WF cannot be explained by the noise of the Kelvin Probe, either
(avg Gradient 301.9 a.u., RMS Gradient 22.2 a.u.). In the photograph (a), neither
scratches nor irregularities are visible to the naked eye, and not even its twin, sample
Cu27, shows such fluctuations.
Figure 6.11 shows, for example, a morphological image of sample Cu28 in comparison
with its WF topography. Even though the sample was polished and coated by the same
company as the best sample (Cu25), it achieved the highest RMS WF value. Its large
fluctuations in WF cannot be explained by the noise of the Kelvin Probe, either. In the
Photograph 6.11a, neither scratches nor irregularities are visible to the naked eye, and not
even its twin, sample Cu27, shows such fluctuations.
To better understand the bad result of sample Cu28 and to see whether the manufac-
turer of samples Cu25 to Cu28 can reproduce the high quality of samples Cu25 and Cu26,
we ordered twins of Cu25 and Cu26 as well as of Cu27 and Cu28. So far, only samples
Cu53 and Cu54 (twins of Cu25 and Cu26) have been investigated, cf. Fn. 7. The RMS
WF values of the new twins are reproducible to ±7.3meV, but the quadruplet exceeds
±25meV. Possible explanations for the bad reproducibility of the pairs of twins are the
poor quality of the polishing (see also the following section) of the new twins and the
fact that the new samples were stored in zip lock bags for about nine months. In the
latter case, the samples might have been corroded by means of perspiration [260] (see also
Sec. 6.3.6). On the other hand, no impurities were found using a SEM analysis [257].
However, the manufacturer of the best gold-plated result is able to reproduce the high
quality of sample Cu12. In comparison with samples Cu3, Cu4, Cu11, Cu32, and Cu33
the RMS WF values are reproducible to ±3.3meV, independent of different processing
techniques and/or surface treatments.
6.3. SAMPLES, DATA, AND RESULTS 185
(a) Full picture (b) Zoom
Figure 6.12: Sample Cu33 (brazed seam). The sample was made of OFHC copper, chopped and
subsequently connected again by means of brazing, polished mechanically and then
by hand, and finally electrolytically coated with 10µm of silver and then 1µm of gold.
(a) Morphological image. (b) Zoom to the brazed seam. Owing to the mechanical
polishing the sample could not be polished to a mirror finish anymore. Nevertheless,
it shows a reasonable homogeneity in WF: RMS WF 20.7meV, PTP WF 98.0meV.
However, the imperfect coating along the brazed seam is prone to chemical reactions
later on.
6.3.6 Surface Conditions
Processing Techniques
As discussed earlier in Sec. 6.3.1 processing techniques were investigated. Namely, the
chopping and subsequently brazing of OFHC copper substrates. Figure 6.12 shows a
photograph of sample Cu33. The sample was chopped and subsequently connected again
by means of brazing, polished mechanically and then by hand, and finally electrolytically
coated. Owing to the mechanical polishing the sample could not be polished to a mirror
finish anymore. Nevertheless, it shows a reasonable homogeneity in WF. Additionally, the
RMS WF values of the twins Cu32 and Cu33 are reproducible to ±3.2meV, cf. Table 6.4.
In comparison with the twins Cu3 and Cu4 (neither chopped nor polished by hand), the
chopping and subsequently brazing seem not to have an influence on the good quality of
the samples. However, the imperfect coating along the brazed seam is prone to chemical
reactions, can trap cleaning solvents, and can become a source of virtual vacuum leaks
later on. We also note that the brazed seam of samples Cu 32 and Cu33 is responsible for
their high RMS grad values.
Surface Treatment
To increase the coating adhesion and, simultaneously, avoid an anisotropic WF, polishing
techniques were investigated. Where possible, the substrate surface was polished mechan-
ically and/or by hand. Figures 6.7a, 6.11a, and 6.12 show examples for hand-polished and
mechanically polished samples. Our investigation revealed that first mechanically polish-
ing excluded polishing to a mirror finish, even after subsequently hand-polishing. The
comparison of the twins Cu11 and Cu12 (hand-polished to a mirror finish) with the twins
Cu3 and Cu4 (mechanically polished) shows that the polishing seems to have an
186 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Table 6.5: Surface roughness (SR) of Kelvin Probe samples. For the hand-polished samples, the
SR profiles were taken twice, once parallel and a second time perpendicular to the
direction of the polishing. The first column gives the name of the sample. The second
column specifies the perthometer parameters Lt (traversing length) and hence Lc (cut-
off; Lc = 0.8mm for Lt = 5.6mm respectively Lc = 0.25mm for Lt = 1.75mm). The
third to fifth column list the perthometer data parallel to the polishing direction; the
sixth to eighth column perpendicular to the direction of polishing. The last column
gives a short comment, if appropriate. All abbreviations are standard nomenclature for
chemical elements. In addition, the following abbreviations are used: Ra (roughness
average), Rmax (maximum roughness depth), and Pc (peak count); see also Fig. 6.13a.
min max
Ref.no. Lt Ra Rmax Pc Ra Rmax Pc Comment
[mm] [nm] [µm] [cm−1] [nm] [µm] [cm−1]
CuRef 5.60 209 2.3 5.0 not polished,
one scan only
Cu41 1.75 301 1.8 64.0 340 2.3 40.0 coated w/ TiN & Ni,
average of 3 scans
Cu39 1.75 12 0.1 0.0 27 2.0 0.0 coated with TiN,
average of 3 scans
Cu40 5.60 89 1.0 0.0 116 1.9 5.0 coated with TiN
Cu42 5.60 37 1.5 0.0 44 2.0 0.0 coated with TiN
Cu47 5.60 71 1.6 0.0 240 5.6 2.5
Cu48 5.60 47 0.8 0.0 159 2.5 0.0
Cu49 5.60 14 <0.1 0.0 49 0.7 0.0
Cu50 5.60 72 1.5 2.5 174 2.7 2.5
Cu51 5.60 14 0.3 0.0 46 0.3 0.0
Cu52 5.60 11 0.1 0.0 89 1.7 2.5
Cu57 5.60 55 1.0 0.0 82 0.9 0.0
Cu58 5.60 19 0.4 0.0 51 0.4 0.0
TiRef 5.60 436 5.2 85.0 not polished,
one scan only
Ti3 5.60 123 1.4 2.5 326 7.3 7.5 coated with TiN
Ti4 5.60 174 1.5 5.0 208 3.4 2.5 coated with TiN
Ti5 5.60 217 2.5 2.5 241 5.5 2.5 coated with TiN
Ti6 5.60 148 2.8 0.0 244 2.9 5.0 coated with TiN
Ti7 5.60 194 3.3 5.0 208 1.8 5.0
Ti8 5.60 159 1.9 2.5 161 1.6 0.0
6.3. SAMPLES, DATA, AND RESULTS 187
influence on the high quality of the samples Cu11 and Cu12: The RMSWF values decrease
from (17.8± 4.6)meV to (12.6± 2.3)meV, i.e., by about 5meV, possibly due to the hand-
polishing. We note that the SR and hence the polishing of the samples determine the
precision of our WF measurements, as discussed in Sec. 6.3.3 (see also Fig. 6.8).
In Table 6.5 we list all the samples studied with a Mahr perthometer H1, a surface
measuring instrument. We note that this is a destructive technique as the perthometer
leaves 5mm long scratches on the sample surface. Figures 6.13 and 6.14 show examples
for SR profile measurements; the perthometer principles and parameters18 are illustrated
in Fig. 6.13a (see also [261]). We note that the investigations with the perthometer started
in the final phase of this thesis. Hence, measurement data exist only for the hand-polished
samples Cu39 to Cu 42 (coated with TiN), Cu47 to Cu52, Cu57 and Cu58, Ti3 to Ti6
(coated with TiN), and Ti7 and Ti8, as well as for the two (not polished) reference samples
CuRef and TiRef.
As stated earlier in Sec. 6.3.1, the unpolished copper and titanium substrates have a
SR better then 210 and 400 nm Ra plus 2.5 and 5.5µm Rmax, respectively. Thanks to the
hand-polishing, the SR of the samples was much improved, although the titanium samples
remain far inferior to the copper samples. The present investigation showed that a sputter
coated sample (Cu39) achieved the lowest SR values. The sample was made of OFHC
copper, hand-polished to a mirror-finish, and sputter coated with 0.8µm of titanium
nitride. Although the Rmax value is in excess of 2µm, the Ra value is (20.0 ± 7.5) nm.
The Ra value of the only hand-polished sample Cu51 was only a few nm higher at (30.0±
22.6) nm. We note that the measured SR values strongly depend on the orientation of the
perthometer relative to the polishing direction, cf. Table 6.5 and Figures 6.13 and 6.14.
For further investigations, we therefore recommend to turn the substrate by 90° after
each polishing step. On average, the Ra values of the copper samples were improved to
(74.6 ± 64.4) nm before and (49.4 ± 38.5) nm after sputter coating; the Ra values of the
titanium samples were improved to (180.5 ± 24.4) nm before and (211.0 ± 63.6) nm after
sputter coating. Unfortunately, a direct comparison with the respective WF values is not
possible.
Layer Thickness
As mentioned earlier, the WF may vary over the surface of an alloy due to non-uniform
segregation of the elements [247]. Plated surfaces may also show corrosion in perspiration
or in sulfur dioxide-containing atmospheres. In Ref. [260] the important influence of inter-
mediate layers on the corrosion behavior of platinized samples was investigated, excerpts
shown in Fig. 6.15. To ensure that the surface coatings provide entire coverage, and to
prevent corrosion and alloy formation on the sample surfaces, various metal combinations
and layer thicknesses19 were investigated. We note that the SR and hence the WF may
18The traversing length Lt is the overall length traveled by the stylus when acquiring the traced profile.
Pre-travel and post-travel are required for phase correct filtering. The values of the surface parameters
are determined over the evaluation length Ln, which comprises five consecutive sampling lengths Li. The
single roughness depth Rzi is the vertical distance between the highest peak and the deepest valley within
a sampling length. The maximum roughness depth Rmax is the largest single roughness depth. The
roughness average Ra is the arithmetic average of the absolute values of the roughness profile ordinates.
The peak count Pc is the number of roughness profile elements per cm which consecutively intersect the
specified upper and lower profile section levels c1 and c2, respectively. In addition, the cut-off of a profile
filter, Lc, determines which wavelengths belong to roughness and which ones to waviness; see also [261]
19In some cases, e.g., in the case of titanium nitride, the layer thickness is limited by the manufacturing
process.
188 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a)
(b)
(c)
(d)
Figure 6.13: Surface roughness (SR) profiles Z(x) of copper samples. (a) Not polished (CuRef),
(b) and (c) hand-polished to a mirror finish (Cu58), and (d) finally sputter coated
with 0.8µm of titanium nitride (TiN) and then electrolytically coated with 1µm of
nickel (Cu41). Please note the different x- and y-scales of the SR profiles. Samples
Cu41 and Cu58 were analyzed (b) parallel and (c) and (d) perpendicular to the
polishing direction. Thanks to the hand-polishing, the SR was improved by a factor
of about 11 (4) parallel (perpendicular) to the polishing direction; the peak count,
Pc, was also reduced to zero. The higher SR in (c) in comparison with (b) is due to
the ripple formation in polishing; the even higher SR in (d) in comparison with (a)
to (c) is due to the miserable adhesion of Ni on TiN. The perthometer principles are
illustrated in (a); its parameters are introduced in Fn. 18
6.3. SAMPLES, DATA, AND RESULTS 189
(a)
(b)
(c)
Figure 6.14: Surface roughness (SR) profiles of titanium samples. (a) Not polished (TiRef), and
(b) and (c) hand-polished to a mirror finish (Ti6). The SR profile of sample Ti6 was
taken twice, (b) once parallel and a second time (c) perpendicular to the polishing
direction. Thanks to the hand-polishing, the SR was improved by a factor of about 3
(2) parallel (perpendicular) to the direction of the polishing; the peak count, Pc, was
also reduced to zero (by a factor of 17). The higher SR in (c) in comparison with (b)
is due to the ripple formation in polishing.
depend on the thickness of the deposited layers [262]. In Refs. [263, 264] is has been found
that the SR increases monotonically with the thickness (of platinum and gold thin films
deposited on silicon and glass substrate). In contrast, in Ref. [265] it has been found
that the SR exhibits a minimum at a certain layer thickness (of silver deposited on glass
substrate). We further note that the degradation under thermal annealing may influence
the SR [263, 266, 267]. According to Table 6.4 the WF might exhibit a minimum at a layer
thickness of 2µm of gold; but we do not have enough data to support this hypothesis.
However, the present investigation showed that a layer combination of 10µm of silver and
then 1µm of gold achieved the lowest and most reproducible RMS WF values.
190 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) Artificial perspiration (b) Sulfur dioxide-containing atmosphere
Figure 6.15: The corrosion behavior of plated samples, for various metal combinations, obtained
after (a) 48 hours in artificial perspiration and (b) 5 cycles in sulfur dioxide-containing
atmospheres. The summary of corrosion data (a) showed the benefit of imposing a
thin layer of gold, silver, or copper-tin between the platinum coating and the copper
substrate. Similar results were obtained after 48 hours salt spray test. Corrosion
data (b) showed some similarities with (a), although a silver intermediate layer was
not as effective unless overlaid with gold. Figures taken from Ref. [260].
6.3.7 Sputtered Gold and SEM Analysis
Sputter coated gold surfaces are reported to give better WF results [240]. Hence, we
have investigated sputtered surfaces, for various metal combinations. We note that these
investigations started in a later phase of this thesis. At that time, the ion source at
University of Mainz, normally used to clean the substrate surface before sputtering, was
inoperative. To get an idea, how important the cleaning process is, only half of the copper
samples were cleaned with isopropyl alcohol before sputtering.
So far, only 8 sputtered samples were analyzed. The sputtered copper samples were
made of OFHC copper, hand-polished to a mirror finish, and sputter coated with 2µm
of silver and then 0.2µm of gold. Figure 6.16 shows WF topographies of copper samples
Cu7 and Cu8, where only sample Cu8 was cleaned with isopropyl alcohol. Both samples
lack the desirable homogeneity in WF. They show a gradual increase in WF from left to
right, only interrupted by sharp peaks, and independent of the cleaning process. Possible
explanations for the poor quality of the sputtered surfaces are an insufficient adhesion,
because of the missing cleaning with the ion source, and the orientation of the sputter
head relative to the sample surface. Indeed, compressed air dusting later destroyed half
of the coating of sample Cu7.
The question, why our sputtered samples lack the desirable quality, was investigated
using a scanning electron microscope (SEM) together with an energy dispersive X-ray
(EDX) analysis20. Figure 6.17 shows SEM images of copper sample Cu22, for different
magnifications. The defects (depression, chunks) shown in Fig. 6.17 lead to large fluctua-
tions in the WF (PTP WF 200meV). For further investigations, we therefore recommend
to shield the sputter head in such a way that meteor-like impacts cannot destroy the WF
homogeneity any more.
20These measurements were taken in cooperation with R. White, the lab manager of the Nano Materials
Characterization Facility (NMCF) at the University of Virginia.
6.3. SAMPLES, DATA, AND RESULTS 191
(a) Not cleaned (b) Cleaned with isopropyl alcohol
Figure 6.16: Samples Cu7 and Cu8 (sputtered gold). The samples were made of OFHC copper,
hand-polished to a mirror finish, and sputter coated with 2µm of silver and then
0.2µm of gold. Work function (WF) topographies, with 25meV contour graduations,
of (a) sample Cu7 not cleaned and (b) sample Cu8 cleaned with isopropyl alcohol.
The samples show a gradual increase in WF from left to right, only interrupted by
sharp peaks, and independent of the cleaning process. One reason for this might be
an insufficient adhesion, due to the missing cleaning with the ion source. Indeed,
compressed air dusting later destroyed half of the coating of sample Cu7.
(a) Magnification factor 450x (b) Magnification factor 1600x (c) Magnification factor 3500x
Figure 6.17: Sample Cu22 (sputtered gold). The sample was made of OFHC copper, hand-polished
to a mirror finish, and sputter coated with 2µm of silver and then 0.2µm of gold,
but not cleaned with isopropyl alcohol before sputtering. (a) Surface at low mag-
nification. (b) Surface at higher magnification, showing depression. (c) Surface at
high magnification, examining chunks. These defects lead to large fluctuations in the
work function (PTP WF 200meV). SEM images courtesy of S. McGovern [243]
In general, the copper and titanium substrates were far inferior to electrolytically
coated samples and achieved RMS WF values of 32meV or more. However, both glass
samples showed reasonable WF values. The sputtered silver sample achieved a RMS WF
value of 8.8meV, competitive with the electrolytically coated samples. The sputtered gold
sample was a few meV higher at 17.1meV. We suspect that the lower SR and the higher
cleanliness of the glass substrates favored lower WF values.
192 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
6.4 Influence of the Patch Effect on a
As discussed earlier in Sec. 3.1.3, the potential difference between DV and AP, UA−U0, has
to be known precisely to determine the transmission function Eq. (3.30) and subsequently
the neutrino-electron correlation coefficient a. However, a variation of the WF within an
electrode or between different electrodes could render the potential difference UA − U0.
Hence, we have to study the influence of our measured WF values on the transmission
function Eq. (3.30). We consider two different scenarios:
Present thesis: The (present) inner surfaces of our electrode system are best described
by the cylindrical sample electrode shown in Fig. 6.1a and samples Cu3, Cu4, Cu32,
and Cu33. In average, the PTP WF of these samples is (133.3± 70.2)meV. On the
one hand, it is very likely that the cylindrical sample electrode overestimates the
WF inhomogeneities, as the DV and AP electrodes e3−e6 and e14 were renewed
respectively over-coated for our latest beam time (see also Fn. 8). On the other
hand, the samples Cu3, Cu4, Cu32, and Cu33 might underestimate the WF inho-
mogeneities, as our electrodes have an unfolded surface of up to 440 × 540mm2,
whereas our samples were only 50× 50mm2 in surface. Altogether, we assume WF
inhomogeneities of 150meV over patches with a surface of 40× 40mm2.
Further investigations: The present investigation showed that a platinized sample
(Cu25) achieved the lowest PTP WF value of 49.0meV, cf. Table 6.2. Hence, we
assume WF inhomogeneities of 50meV over patches with a surface of 20× 20mm2.
Figures 6.18 to 6.21 show the possible influence of the patch effect on the electric field
distributions in the AP and in the DV, respectively.
6.4.1 Patch Effect in the Analyzing Plane
Assuming that the DV is grounded (U0 ≡ 0), we have to know the potential barrier UA
with an accuracy of better than 10mV, in order to keep systematic uncertainties in a
below ∆a/a = 0.1% (for details see Sec. 3.1.3 and Fig. 3.6). Indeed, a variation of the
WF within the AP electrode e14 could render the electric field distribution. Figure 6.18
shows the influence of one patch on the electric potential in the AP. Only in the case of the
present thesis, the electrostatic potential exceeds UA + 10mV on the right detector pad.
The left and central detector pads are almost not affected by the WF inhomogeneities.
For the central detector pad, the field inhomogeneity is in excess of 10mV only on the
right boundary.
In case of Fig. 6.18, the electric field calculations showed that the field inhomogeneity
• averaged over the central detector pad is around 5mV for the present situation, what
corresponds to a shift in a of ∆a/a = +0.06(5)%,
• over the central detector pad is less than 1.1mV for further investigations, which
yields a shift in a of ∆a/a = +0.01(5)%. Similar results were obtained for the
left and right detector pads. The electric field inhomogeneity averaged over the
right detector pad is around 3mV, what corresponds to a shift in a of +0.04(5)%.
However, for the left detector pad, the field inhomogeneity is less than 0.02mV,
which yields a shift in a of less than 0.01(5)%.
6.4. INFLUENCE OF THE PATCH EFFECT ON A 193
(a) (b)
(c) (d)
Figure 6.18: Influence of work function (WF) inhomogeneities on the electric potential in the
analyzing plane (AP) at z = 1.32m, for (a) and (c) the present thesis and (b) and
(d) further investigations, with UA = 50V. We mention that almost the same results
were obtained for, e.g., UA = 400V. (a) and (b) show the electric field distribution.
Only in the case of the present thesis (a), the electrostatic potential exceeds 50.01V on
the right detector pad. Please note the different contour scales. The left and central
detector pads are almost not affected by the WF inhomogeneities. Here, the black
lines represent the electrodes e1 to e16, the two heat shields for the detector electrode
e17, as well as the vacuum tube. (c) and (d) show the electric field inhomogeneity,
U(x)−UA, after deduction of the barrier voltage UA, for 4 different widths y. For the
central detector pad, the field inhomogeneity is in excess of 10mV only in the case
(c) and on the right boundary. The gray bar represents the 10mV boundary. Please
note the different y-scales. For elucidation, the golden frames show a projection of
the (proton) detector chip into the AP; the red frames (lines) include two gyration
radii on each side (for B0 = 2.177T). Here, we neglect the spatial displacement of
both the detector chip and by the lower and upper dipole electrodes.
We note that Fig. 6.18 is only an example for both possible patches in the AP and the
influence of the patch effect on the angular correlation coefficient a. Figure 6.19 shows
further examples for the influence of the patch effect on the electric field distribution in
the AP. In both cases, all three detector pads are affected by the WF inhomogeneities.
The electric field inhomogeneity is in excess of 10mV over the entire detector chip. In the
194 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.19: Influence of work function inhomogeneities on the electric potential in the analyzing
plane (AP) at z = 1.32m, for the present thesis. With (a) 3 patches, the electric field
inhomogeneity is already in excess of 10mV over the entire detector chip; with (b)
5 patches, the field inhomogeneity averaged over the central detector pad is around
27mV, what corresponds to a shift in a of ∆a/a = +0.37(7)%. The (present) inner
surface of the AP electrode is best described by (b), where the five patches correspond
to one brazed seam of the AP cylinder and four welding points for the supporting
arms of the AP. Here, the black lines represent the electrodes e1 to e16, the two heat
shields for the detector electrode e17, as well as the vacuum tube.
case of 5 patches (Fig. 6.19b), the field inhomogeneity averaged over the central detector
pad is around 27mV, what corresponds to a shift in a of
∆a/a = +0.37(7)%. (6.9)
We note that the (present) inner surface of the AP electrode is best described by Fig. 6.19b.
Thus, for the present thesis, the influence of the patch effect in the AP on the neutrino-
electron correlation coefficient a is negligible at the level of 1%. However, for further
investigations, the influence of the patch effect in the AP on a is negligible at the level of
0.1%, as electric field calculations of, e.g., 8 patches in the AP show.
6.4.2 Patch Effect in the Decay Volume
In our measurement, the DV is grounded. We therefore can assume that U0 ≡ 0. On the
other hand, a variation of the WF within the DV electrode gr could render the electric
field distribution. Figure 6.20 shows the influence of one patch on the electric potential
in the DV. Only in the case of the present thesis, the electrostatic potential exceeds
10mV, but almost over the entire cross-section of the neutron beam. However, for further
investigations, the electrostatic potential is well below 10mV.
In the case of Fig. 6.20, the electric field calculations showed that the field inhomo-
geneity averaged over the the central detector pad
• is around 20mV for the present situation, what corresponds to a shift in a of
∆a/a = −0.24(5)%, (6.10)
6.4. INFLUENCE OF THE PATCH EFFECT ON A 195
(a) (b)
(c) (d)
Figure 6.20: Influence of work function inhomogeneities on the electric potential in the de-
cay volume (DV) at x = 0, for (a) and (c) the present thesis and (b) and (d)
further investigations, with U0 ≡ 0V. (a) and (b) show the electric field distri-
bution. Only in the case of the present thesis (a), the electrostatic potential
exceeds 10mV, but almost over the entire cross-section of the neutron beam.
Please note the different contour scales. For comparison, the black contour
lines indicate the neutron beam profile. (c) and (d) show the electrostatic po-
tential, for 4 different heights z. In the case (c), the electrostatic potential
exceeds 10mV over the entire width of the (proton) detector, for all 4 heights.
However, in the case (d), the electrostatic potential is well below 10mV. The
gray bar represents the 10mV boundary. Please note the different y-scales.
For elucidation, the red lines show a projection of the detector into the DV,
including two gyration radii on each side (for B0 = 2.177T). Here, we neglect
the spatial displacement of both the detector chip and by the lower and upper
dipole electrodes.
196 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.21: Influence of work function (WF) inhomogeneities on the electric potential in the
decay volume (DV) at (a) x = 0 and (b) x = −11mm, for the present thesis. With a
variation of the WF over (a) the rear side part (of the side port) of the DV electrode
gr, the electrostatic potential exceeds 10mV over almost the left half of the neutron
beam; with a variation of the WF over (b) the upper ring of the DV electrode gr,
the electric potential exceeds 10mV only over the top quarter of the neutron beam.
However, in the case of (b), decay protons may be trapped between the DV and
the electrostatic mirror, as the electric field distribution exhibits a potential barrier
above the neutron beam, of up to 105mV at the edges of the detector chip. Such a
variation of the WF reflects the possible influence of the brazed seam on top of the
DV electrode gr. See the text for details. Here, the black lines represent the DV
electrodes gr and e6.
Similar results were obtained for the left and right detector pads. The electric field
inhomogeneity averaged over an outer detector pad is around 15mV, which yields a
shift in a of −0.18(5)%.
• is around 2mV for further investigations, what corresponds to a shift in a of
∆a/a = −0.02(5)%. (6.11)
Similar results were obtained for the left and right detector pads. The electric field
inhomogeneity averaged over an outer detector pad is around 1.5mV, which yields
a shift in a of less than -0.02(5)%.
As in the case of the AP, Fig. 6.20 is only an example for both possible patches in
the DV and the influence of the patch effect on the angular correlation coefficient a.
Figure 6.21 shows additional examples for the influence of the patch effect on the electric
field distribution in the DV. In both cases, only a part of the neutron beam is affected by
the WF inhomogeneities. Thus, Eq. 6.10 sets an upper limit for the influence of the patch
effect in the DV on the angular correlation coefficient a.
However, in the case of Fig. 6.21b, the electric field distribution exhibits a potential
barrier above the neutron beam, of up to 105mV at the edges of the detector chip. We
6.4. INFLUENCE OF THE PATCH EFFECT ON A 197
mention that such a variation of the WF reflects the possible influence of the brazed
seam on top of the DV electrode gr. As a result, decay protons may be reflected at
the potentail barrier and subsequently trapped between the DV and the electrostatic
mirror. Similarly, in the case of Figs. 6.20a to 6.21a, the electric field distribution exhibits
a potential barrier in the DV, at, e.g., z = 0. Depending on their initial momentum,
protons emitted below this potential barrier (at negative z-positions) may be reflected at
the maximum and trapped between the DV and the electrostatic mirror. This, in turn, will
lead to unexpected background due to interactions of the trapped protons with the residual
gas and/or other decay products. For the same reason, local magnetic field maxima in
the DV must be avoided. For protons emitted above the potential barrier (at positive z-
positions), the condition for adiabatic transport may be violated, also depending on their
initial momentum. Both effects will lead to a distortion of the adiabatic transmission
function Eq. (3.30). Hence, the influence of the patch effect in the DV on the neutrino-
electron correlation coefficient a is not negligible.
6.4.3 Unexpected Proton Reflections and Influence on a
In Refs. [5, 33, 46, 47, 50] we did not consider possible proton reflections from the DV.
To incorporate the patch effect and hence the proton reflections into the transmission
function Eq.(3.30), we make the following Ansatz:
WF inhomogeneities in the DV are described by an electric field gra-
dient ∂U0/∂z > 0, between the decay point and the potential barrier.
This electric field gradient will be superimposed on the slight magnetic field gradient
∂B0/∂z < 0 in the DV, cf. Sec. 3.2.2. The slight magnetic field gradient, in turn, causes a
slow and adiabatic momentum transfer from transverse to longitudinal motion. On their
way from the decay point P0 to the potential barrier PDV, the polar angle of the protons
will therefore have changed from its initial value θ0, i.e., |90°−θ0| will have increased. Thus,
the magnetic field gradient weakens the influence of the patch effect. In the following, we
neglect the possible effect of non-adiabatic proton motion. This could be part of further
investigations (by means of MC simulations).
The potential barrier in the DV, caused by the patch effect, acts as an electric mirror.
For a given definite kinetic energy T0 only protons with sufficient longitudinal momen-
tum can overcome this electric mirror. In the adiabatic approximation, the condition for
transmission through the (po(tential barrier)can be)written as (cf. Eqs. (3.23) and (3.24)):∂B0
T ad − ∂U(P ) = 0
B0 + ∆z !
|| DV T0 e U
∂z 2
0 + ∆z − U0 − T0 sin θ0 > 0, (6.12)
∂z B0
where ∆z = z(PDV)− z(P0) > 0(. Solving inequation (6.12)) for T0 yields:−1
∂U B ∂B0
( ) = 0 ∆ 1− 0
+ ∂z ∆zT0 > Ttr,DV θ0 e z sin2 θ0 . (6.13)
∂z B0
198 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Hence, protons with21
T > Tmax0 tr,DV =(Ttr,DV(θ0 = ±90°) )
∂B −1 ( ) (6.14)∂U B + 0 ∆z ∂U ∂B −1
= 0 ∆ 1− 0 ∂z = − 0 0e z e B0 = O(100 eV)
∂z B0 ∂z ∂z
are transmitted through the potential barrier, whereas protons with
min ∂UT0 < Ttr,DV = Ttr,DV(θ0 = 0°) =
0
e ∆z = O(10meV) (6.15)
∂z
are reflected at the potential barrier and subsequently trapped, both independent of their
(initial) polar angle. Solving inequation (6.12) for θ0, in turn, yields that, for initial
energies Tmintr,DV < T < T
max
0 tr,DV, protons can pass the potential barrier only if their (initial)
polar angle goes below (se√e also Fn. 3 in C√√√ (
hap. 3) )
B ∂U0max = arcsin 0
e ∂z ∆zθ0 < θtr,DV 1− , (6.16)
B0 + ∂B0∂z ∆z T0
i.e., protons with an initial energy of Tmin maxtr,DV < T0 < Ttr,DV are transmitted through the
potential barrier with ∫a probability of (sθmax [ee also F]n. 4 in Chap. 3)θmax ( √ )tr,DV tr,DV
wtr,DV(T0) =
0 √ dθ sin θ = − cos θ = 1− 1− sin2 θmaxtr,DV√ 0√√ ( )B e∂U0 ∆z= 1− 1− 0 1− ∂z . (6.17)
B + ∂B00 ∂z ∆z T0
Thus, the proton reflections in the DV are described by the transmission function: 0 √ ( ) , T0 ≤ Tmintr,DV∂U0
Ftr,DV(T0) =  1− 1−
B0
∂B 1−
e ∆z
∂z , Tmin max
B + 0 ∆z T0 tr,DV
< T0 ≤ Ttr,DV .
0 ∂z
1 , T max0 > Ttr,DV
(6.18)
To finally calculate the influence of the proton reflections on the angular correlation
coefficient a, we have to convolve the transmission functions Eq. (3.30) with Eq. (6.18).
Depending on the relative figures of Tmin , Tmin, Tmax maxtr,DV tr tr,DV, and Ttr , the transmission
functions can intersect or not. If, however, the two transmission functions have no common
point of intersection, the convolution is one of the two transmission functions Eq. (3.30)
or Eq. (6.18). Altogether, the patch effect in the DV can be described by the transmission
21The orders shown in Eqs. (6.15) and (6.15) are based on the assumptions that ∂U0/∂z =
O(10mVcm−1), (∂B0/∂z) /B0 = O(1× 10−4 cm−1), and ∆z = O(1 cm).
6.4. INFLUENCE OF THE PATCH EFFECT ON A 199
function:
 0 √ ( ) , T0 ≤ T
min
 tr,PE 1−√1− B0 eUA min inter B
1−
A T(0 ) , Ttr,PE < T0 ≤ Ttr,PEFtr,PE(T0) = 
,
∂U
 1− 1− B 1− e
0 ∆z
 0 ∂z , T inter max ∂BB + 0 ∆z T0 tr,PE < T0 ≤ Ttr,PE0 ∂z1 , T > Tmax0 tr,PE
(6.19)
where ( )
Tmintr,PE = Max( Tmin;Tmintr tr,DV( and ) ) (6.20)
Tmax max maxtr,PE = Min Tp,max; Max Ttr ;Ttr,DV . (6.21)
The point of intersection, if existent, is located at the intersection of the second and third
line of Eq. (6.19),i.e., Tma(x ) , Tmin mintr,PE tr,PE < Ttr and Tmax maxtr,PE < Ttr∂B ∂U
inter  eUA B0+ 0 ∆z −e 0 ∆zB=  ∂z ∂z ATtr,PE  .∂B , otherwiseB 00+ ∆z−B∂z A
Tmin , Tmintr,PE tr < T
min
tr,PE and T
max
tr < T
max
tr,PE
(6.22)
We note that the transmission function Eq. (6.19) reduces to the familiar expression
Eq. (3.30) if Tmin < Tmin max maxtr,PE tr and Ttr,PE < Ttr , i.e., in the case of high AP voltages
(UA ≥ 250V) and/or small WF inhomogeneities, as shown in Fig. 6.22b. We further note
that the transmission function will not reach 100% any more if Tmaxtr,DV > Tp,max, i.e., in
the case of large WF inhomogeneities, as shown in Figs. 6.22a and 6.22b.
Figure 6.22 shows the influence of the proton reflections on the transmission function.
The figure shows that the slight magnetic field gradient ∂B0/∂z < 0 in the decay volume
dramatically reduces the influence of the patch effect. In addition, Fig. 6.23 shows the
influence of the patch effect on the angular correlation coefficient a. The higher the WF
inhomogeneities ∂U0/∂z are, the worse the systematic error in a is. For electric field
gradients ∂U0/∂z > 100mVcm−1, the systematic error even exceeds 18%.
In order to calculate the systematic error in a for the present thesis, we first have to
determine the size of the WF inhomogeneities ∂U0/∂z over the central detector pad. In
Table 6.6 we list the electric field gradients ∂U0/∂z for the two extreme cases shown in
Fig. 6.20a and 6.21b. In the case of Fig. 6.20a the electric field distribution exhibits a
maximum at z = 0; in the case of Fig. 6.21b at z = +6 cm. In the direction of the neutron
beam the electric field gradients are nearly independent of the depth x. The closer one
gets in y-direction to the maximum, the higher the electric field gradients and the shorter
their height expansion Lz get.
Next, we have to weight the electric field gradients with the neutron beam distribu-
tion. W.l.o.g. we assume a uniform neutron beam distribution over the central detector
200 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
(a) (b)
Figure 6.22: Influence of work function inhomogeneities on the transmission function, for two
different analyzing plane (AP) voltages: (a) UA = 50V and (b) UA = 250V, with
∆z = 1 cm, B0 = 2.177T, and BA/B0 = 0.203. For better visibility, the y-axis is
enlarged to (95 − 100)%. The comparison between the green and red lines shows
that the slight magnetic field gradient ∂B0/∂z < 0 in the decay volume dramatically
reduces the influence of the patch effect. We mention that for high AP voltages
(UA ≥ 250V) the transmission function Eq. (6.19) reduces to the familiar expression
Eq. (3.30), shown in black, depending on the size of the magnetic field gradient
∂B0/∂z.
pad. Then, the systematic errors in a are approximated with the averaged electric field
gradients. A more sophisticated analysis will be part of further investigations (by means
of MC simulations).
In the case of Fig. 6.20a, only the top half of the neutron beam is affected by the proton
reflections. Thus, for positive z values, we have to weight the electric field gradients with
0. In the event that Lz does not cover the whole bottom part of the neutron beam, we
have to redefine the electr〈ic fi〉eld gradient over the u∫ncovered part of the neutron beam:
∂U L00 7−→ ∂U0 1 ∂U 1 1Lz = 0Lz ∫ dz (6.23)
∂z ∂z z ∂z L0
L −L dzz L0−L
z
z
0 z
∂U0 [ ]1 [ ]L0 ∂U 2 L= 0Lz ln z = ln 0 ,
∂z 1 z2
L0 L0−Lz ∂z 2L0 − Lz L0 − Lz
2 L0−Lz
where L0 denotes the distance between the lower edge of the neutron beam and the
maximum of the electric field (here: z = 0, i.e., L0 ≈ 4 cm). Thus, for Lz < L0, we have
to weight(the electric field gradients with )
1
(L0 −
2 L
Lz) ln
0 + Lz . (6.24)
L0 2L0 − Lz L0 − Lz
This method gets an average electric field gradient of ∂U0/∂z = (2.2 ± 2.9)mVcm−1.
Together with our magnetic field gradient of (∂B0/∂z) /B0 = −1 × 10−4 cm−1 (see also
Fn. 22 in Chap. 3) this corresponds to a systematic error in a of
∆a/a = (0+0.04−0.00 ± 0.04)%. (6.25)
Here, the first error stems from the uncertainty in the electric field gradient and the latter
one from the respective errors shown in Fig. 6.23.
6.4. INFLUENCE OF THE PATCH EFFECT ON A 201
Figure 6.23: Relative change of the angular correlation coefficient a for different work function
(WF) inhomogeneities in the DV, described by an electric field gradient ∂U0/∂z > 0.
The higher theWF inhomogeneities ∂U0/∂z are, the worse the systematic error in a is.
For electric field gradients ∂U0/∂z > 100mVcm−1, the systematic error even exceeds
18%. The comparison between the green, red, and black line shows that the slight
magnetic field gradient ∂B0/∂z < 0 in the decay volume dramatically reduces the in-
fluence of the patch effect. Input data for the calculation (with Nachtmann’s Formula
Eq. (2.38)): ∆z = 1 cm, B0 = 2.177T, BA/B0 = 0.203, UA = 50, 250, 400, 500, 600V,
and the recommended value for a = −0.103 [10].
In the case of Fig. 6.21b, the electric field gradient stretches only over Lz ≤ 6 cm,
i.e., over positive z values. In the event that Lz does not cover the whole top part of
the neutron beam, we have to weight the electric field gradients with Eq. (6.24) (here:
L0 = 6 cm). For negat∫ive z values, we have to weight the electric field gradients with∫ 1 L0 1 2Lz LLz dz = ln 0 , (6.26)L0 dzz L −L z 2L L − L2− 0 max 0 max max L0 − LmaxL0 Lmax
where Lmax denotes the distance between z = 0 and the maximum of the electric field
(here: z = +6 cm, i.e., Lmax ≈ 6 cm and L0 ≈ 10 cm). This way we get an average electric
field gradient of ∂U0/∂z = (4.6± 3.6)mVcm−1, which yields a systematic error in a of
∆a/a = (+0.03+0.32−0.03 ± 0.08)%. (6.27)
Thus, for the present thesis, the influence of the patch effect in the DV on the neutrino-
electron correlation coefficient a is negligible at the level of 1%. However, the effect
of possible proton trapping between the DV and the mirror electrode, in the order of
0.1 s−1 for a full proton decay rate of N0 = 490 s−1, requires further investigations (by
means of MC simulations). However, for further investigations, the influence of the patch
effect in the DV on a is negligible at the level of 0.1%. In the case of Fig. 6.21b, but
with a variation of the WF of only 50mV, we get an average electric field gradient of
∂U0/∂z = (1.5 ± 1.2)mVcm−1. This corresponds to a systematic error in a of only
∆a/a = (0.00+0.01−0.00 ± 0.02)%.
202 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Table 6.6: Work function (WF) inhomogeneities in the DV, described by an electric field gradient
∂U0/∂z. Here, we consider the two extreme cases shown in Fig. 6.20a (columns 2 to
5) and 6.21b (columns 6 to 9). In the case of Fig. 6.20a the electric field distribution
exhibits a maximum at z = 0; in the case of Fig. 6.21b at z = +6 cm. The electric field
gradients stretch over Lz in height from this maxima. In the direction of the neutron
beam we only list the three values x = 0,±11mm, as the electric field gradients are
nearly independent of the depth x. The closer one gets in y-direction to the maximum,
the higher the WF inhomogeneities and the shorter the height expansions Lz get.
Patch (see Fig. 6.20a) Ring (see Fig. 6.21b)
Depth x = 0 x = ±11 mm x = 0 x = ±11 mm
Width y ∂U0/∂z Lz ∂U0/∂z Lz ∂U0/∂z Lz ∂U0/∂z Lz
[mm] [mVcm−1] [cm] [mVcm−1] [cm] [mVcm−1] [cm] [mVcm−1] [cm]
-20 1.5 4.0 1.3 4.0 9.2 6.0 10.0 6.0
-15 2.0 4.0 1.8 4.0 8.3 6.0 8.3 6.0
-10 2.5 4.0 2.3 4.0 8.3 6.0 8.3 6.0
-5 3.5 4.0 3.1 4.0 7.9 6.0 8.3 6.0
0 4.3 4.0 3.8 4.0 7.9 6.0 8.3 6.0
5 6.6 3.5 6.0 3.5 7.9 6.0 8.3 6.0
10 9.1 3.5 8.9 3.5 7.9 6.0 8.3 6.0
15 12.9 3.5 11.4 3.5 8.3 6.0 8.3 6.0
20 22.0 2.5 18.3 3.5 8.3 6.0 9.2 6.0
25 38.5 2.0 37.0 2.0 43.0 2.0 42.5 2.0
30 120.0 1.0 100.0 0.5 110.0 1.0 260.0 0.5
In summary, for the present thesis, the patch effect introduces an unexpected system-
atic error to the angular correlation coefficient a of
∆a/a = (+0.03+1.93−1.64 ± 0.18)% (6.28)
consisting of:
• Proton reflections in the DV: (+0.03+0.32−0.03 ± 0.08)%, from Eq. (6.27),
• Inhomogeneity of the WF in the AP: ±0.37(7)%, from Eq. (6.9),
• Inhomogeneity of the WF in the DV: ±0.24(5)%, from Eq. (6.10), and
• Absolute WF values in the DV and the AP: ±1.00(13)%,
assuming that the absolute WF values in the DV and the AP would differ by about
75mV. This would describe, e.g., different aging processes of the DV and the AP
electrode as observed for the cylindrical sample electrode shown in Fig. 6.1a.
We note that the sign of three of the errors is unknown since we do not know the sign of
the WF inhomogeneities, except for the proton reflections.
6.4. INFLUENCE OF THE PATCH EFFECT ON A 203
Figure 6.24: Electric field distribution in the decay volume (DV) for ±2V on the bottom and top
cylinders e3 and e6 of the DV electrode, respectively, with 20mV contour graduations.
With this setting we can eliminate both local electric field maxima in and above the
DV. Inside the DV the electric field gradient is not less than 25mVcm−1; at the top
end of the DV electrode it is up to 350mVcm−1. Here, the black lines represent the
DV electrodes gr and e6.
6.4.4 Approaches for the Unexpected Proton Reflections
As discussed in the previous section, there are possibilities to reduce the influence of the
patch effect on the neutrino-electron correlation coefficient a. Here, we only list approaches
to eliminate local electric field maxima in the DV and/or to handle the unexpected proton
reflections:
Slight magnetic field gradient: The slight magnetic field gradient in the DV causes
a slow and adiabatic momentum transfer from transverse to longitudinal motion.
Thus, it weakens the influence of the patch effect on a, as shown in Fig. 6.23. We
therefore have to investigate the impact of various magnetic field gradients on a as
well as on the proton count rates, in a further beam time. As stated in the previ-
ous section, the transmission function Eq. (6.19) reduces to the familiar expression
Eq. (3.30) in the case of high AP voltages (UA ≥ 250V), as shown in Fig. 6.22b.
As a first step, it is sufficient to only study the proton count rates at AP voltages
UA = 50V and UA = 250V for different magnetic field gradients. On the other
hand, a too large increase of the magnetic field gradient in the DV goes hand in
hand with the violation of the condition for adiabatic transport. Thus, it is to be
hoped that the proton count rates stabilize at a reasonable size of the magnetic field
gradient.
Inverse electric field gradient: Protons may be trapped between the DV and the elec-
tric mirror only in the event of a positive electric field gradient in the DV or an electric
field maximum above the DV. Consequently, it suggests itself to turn the electric
field gradient in opposite direction, by means of a superimposed external electric
field. The simplest way is to apply a potential of a few Volts to the bottom and
204 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Figure 6.25: Relative change of the angular correlation coefficient a for different work function
(WF) inhomogeneities, ∆U6, in the top cylinder e6 of the DV electrode. The higher
the electrostatic barrier U6 is, the higher the systematic change in a is. But, the
comparison between the green, red, and black line shows that the electrostatic barrier
U6 dramatically reduces the sensitivity of this change on the patch effect. For WF
inhomogeneities ∆U6 ≤ 10mV, the sensitivity is only in the order of permille. Input
data for the calculation (with Nachtmann’s Formula Eq. (2.38)): B0 = 2.177T,
BA/B0 = 0.203, UA = 50, 250, 400, 500, 600V, and the recommended value for a =
−0.103 [10].
top cylinders e3 and e6 of the DV electrode, with opposite sign. Figure 6.24 shows
the influence of ±2V on the electric field distribution in the DV. With this setting
we can eliminate both local electric field maxima in and above the DV. Inside the
DV the electric field gradient ∂U0/∂z is not less than 25mVcm−1; at the top end
of the DV electrode it is up to 350mVcm−1. Compared with Figs. 6.20, 6.21, and
Table 6.6 these gradients might already be too strong. As in the first approach, we
have to study the influence of different electric field gradients on a as well as on the
proton count rates, in a further beam time. We mention that one such first test in
our latest beam time at the Institut Laue-Langevin (ILL) was inconclusive, due to
non-statistical fluctuations of the count rates (for details see Sec. 5.3.3). As in the
case of the magnetic field gradient, a too large increase of the electric field gradient
is accompanied with the violation of the condition for adiabatic transport.
Definite electric mirror above the DV: Instead of compensating local electric field
maxima, we can introduce a small but known potential barrier above the DV. For
instance, we can apply a potential of a few Volts to the top cylinder e6 of the DV
electrode. This causes a systematic but correctable change in a of several 10%,
as can be seen from Fig. 6.25. Fortunately, the sensitivity of this change on WF
inhomogeneities is only in the order of permille. A great disadvantage of this method
is that the potential barrier will be located at around z = +13 cm, depending on the
settings of the lower E×B electrode e8, far above the DV. At this height, neither the
magnetic nor the electric field will be as homogeneous as in the DV or the AP. Thus,
6.5. SUMMARY AND OUTLOOK 205
we have to analyze whether the potential barrier can be described by a transmission
function similar to the familiar one Eq. (3.30) or Eq. (6.19). This will be part of
further investigations (by means of MC simulations).
Definite magnetic mirror above the DV: Instead of a potential barrier above the
DV, we can introduce a small but known magnetic mirror (cf. also the measure-
ment principles of PERC in App. C.1). For instance, we can inverse the magnetic
field gradient so that we shift the local magnetic field maximum from z = −6 cm at
present to, e.g., z = +6 cm. A great advantage of this method is that the magnetic
mirror will be located at a rather homogeneous magnetic field. However, we have to
find a setting of the correction coils c3 and c5 which compensates the proton trap-
ping due to variations of the WF and simultaneously does not violate the condition
for adiabatic transport. This has to be part of further investigations (by means of
MC simulations).
We note that in the last two proposals we must pay attention to the trapped protons.
Possible ways to clean the proton trap between mirror electrode and potential barrier or
magnetic mirror are:
• introducing a further E×B electrode between DV and electric mirror. For instance,
the quadrupole electrode e2 may be connected like an E ×B electrode. The great
disadvantage of this method is that the magnetic field at height of the quadrupole
lacks the desirable homogeneity. A violation of the condition for adiabatic transport
may be the result.
• periodical switching on and off of the electric mirror, or
• measurements in which the electric mirror is switched off completely.
6.5 Summary and Outlook
In order to keep systematic uncertainties in the neutrino-electron correlation coefficient
a below ∆a/a = 0.1%, the potential difference between DV and AP has to be known
precisely. Indeed, a variation of the WF within an electrode or between different electrodes
could render this potential difference. Hence, we investigated experimentally the patch
effect and theoretically its impact on a.
6.5.1 Investigations of the Patch Effect and Influence on a
The present investigation showed that a platinized sample (Cu25) achieved the lowest WF
values: RMS WF 6.6meV and PTP WF 49.0meV. The WF values of its twin (Cu26) were
only a few meV higher at RMS WF 9.0meV and PTP WF 54.0meV. Although the RMS
WF value of sample Cu25 was stable to ±2.2meV over months, the manufacturer of this
sample was not able to reproduce its good quality. However, a gold-plated sample (Cu12)
achieved the lowest and most reproducible WF values: RMS WF 10.9meV, PTP WF
54.0meV, stable to ±3.0meV over at least 9 months, and reproducible to ±3.3meV, inde-
pendent of different processing techniques and/or surface treatments. Our investigations
with the SKP have shown that coating adhesion and SR both have a significant influence
on the measured WF values. In particular, our analysis suggests that hand-polishing to a
mirror finish decreases the RMS WF values by about 5meV.
206 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Platinum has the highest polycrystalline WF value, as shown in Table 6.1, nearly
200meV greater than the highest mono-crystalline WF value of gold. Low WF values,
in turn, are prone to oxidation. This would speak for platinum as outer surface coating,
without experimental proof. However, the WF data presented in this thesis are not clearly
in favor of platinum or gold. With regard to the reproducibility of the samples, we suggest
gold as outer surface coating for the upcoming measurements with aSPECT.
Nevertheless, a few questions remain:
• Whether and if so to which electrode size the present results can be upscaled?
• Why could the sample with the lowest WF values not be reproduced?
• Whether and if so how would rolling or bending alter the WF?
• Whether and if so how much could an UHV compatible SKP improve the measured
WF values? (e.g., in comparison with Ref. [240])
Therefore, for further investigations, we suggest to not only scan the samples with the
SKP but also
• study the SR, the coating composition, and the sample surface by means of a
perthometer respectively a SEM together with an EDX analysis, and
• superimpose the SR, SEM, and EDX results on the measured WF data,
in order to draw significant conclusions.
For the latest beam time, we have shown that the patch effect introduces an unexpected
systematic error to the angular correlation coefficient a of
∆a/a = (+0.03+1.93−1.64 ± 0.18)%, (6.29)
caused by proton reflections in the DV, WF inhomogeneities in the AP, WF inhomo-
geneities in the DV, and different absolute WF values in the DV and the AP. However,
for an improved electrode system we expect that the influence of the patch effect on a is
negligible at the level of 0.1%.
6.5.2 Improvements for a Further Beam Time
Based on our investigations with the SKP, we suggest, for further beam times:
• avoid welding and brazing, if not possible prefer brazing, otherwise plugging into
each other or screwing,
• favor flat shapes over cylinders for the electrodes, e.g., construct the AP electrode,
shaped as a regular polygon, out of small plates,
• polish the substrates at least by hand to a mirror finish, turn the substrates by 90°
after each polishing step,
• clean the surfaces thoroughly from remnants of the polishing compound,
6.5. SUMMARY AND OUTLOOK 207
and in particular for the upcoming measurements:
• re-open the right side of the DV electrode gr, to reduce the influence of WF inho-
mogeneities as shown in Fig. 6.20 on a and simultaneously increase the pumping
speed,
• increase the volume of the DV and subsequently the distance between the neutron
beam and the inner surfaces of the DV electrode gr,
• use a layer combination of first 10µm of silver and then 1µm of gold.
We mention that there are a plenty of reasons for an in-situ measurement of the WF,
by means of an UHV SKP periodically switching between DV and AP:
• different absolute WF values in the DV and the AP,
• adsorption of residual gas on the inner surfaces of the electrodes,
• aging processes of the electrodes, and many more.
6.5.3 Upcoming Measurements with aSPECT
Currently, the aSPECT experiment is set up at the PF1B of the ILL. With measure-
ments in the June/July and the September/October 2011 cycle, we aim to improve the
uncertainty in a to 1%, well below the present literature value of 4% [10].
For this purpose, the DV electrodes e3, gr, and e6 were already renewed. In view of
the short time till the beam time, spare parts for the DV electrode gr were produced to
be analyzed with the SKP. So far, the two spare parts T1 and T2 (plates, 55 × 110mm2
in surface) were analyzed. The SR analysis yielded Ra values of 48(19) nm and 40(19) nm,
respectively [268]. With the SKP, RMS WF values of (28.8 ± 11.1)meV and (33.0 ±
4.7)meV, respectively, were achieved [257]. Both values are comparable with but inferior
to our samples Cu3, Cu4, Cu11, Cu12, Cu32, and Cu33. Thus, we expect that the impact
of the patch effect on the angular correlation coefficient a is negligible at the level of 0.1%,
except for different absolute WF values in the DV and the AP.
208 CHAPTER 6. INVESTIGATIONS OF THE PATCH EFFECT
Chapter 7
Summary and Outlook
The retardation spectrometer aSPECT [46, 47] has been built to perform precise mea-
surements of the antineutrino-electron angular correlation coefficient a, by measuring the
proton recoil spectrum in the decay of free, unpolarized neutrons. A precise measurement
of a can be used to study the Standard Model of elementary particles and fields as swell
to search for evidence of possible extensions to it, like right-handed currents or scalar and
tensor interactions, cf. Secs. 2.2 and 2.3.
7.1 Statistical and Systematic Limits
In this thesis the latest measurement with the aSPECT spectrometer at the Institut Laue-
Langevin (ILL) in Grenoble, France (2007-2008), is described. The primary focus of this
beam time was on the identification and investigation of possible systematic effects. The
overall aim of the aSPECT experiment is to improve the accuracy of the neutrino-electron
correlation coefficient a by more than one order of magnitude in comparison to previous
measurements [40, 41]. In the latest beam time a statistical accuracy of about 2% per
24 hours measurement time was reached.
Compared with our first beam time at the Forschungs-Neutronenquelle Heinz Maier-
Leibnitz in Munich, Germany (2005-2006) [33], the major improvements for this beam
time were a new proton detector, the redesign of several electrodes (cf. Sec. 3.2.1), and
improved ultra high vacuum. For the first time, a silicon drift detector was used [52]. This
detector has a much better separation of signal and noise. With the new detector, the
acceleration potential could be significantly reduced and therefore the problem of frequent
electrical breakdowns be solved [34].
During the data analysis several different systematic effects were investigated. Ex-
haustive Monte Carlo simulations were performed to compute a variety of systematic
corrections, cf. Sec. 5.5. However, the data analysis revealed a problem in the detector
electronics which caused a significant systematic error. Unfortunately, this effect was dis-
covered only after the beam time. Despite our best efforts, we could only set upper limits
on the correction of the problem, which are too high to determine a meaningful result
from our latest beam time at the ILL, cf. Sec. 5.4.1.
Apart from the problem in the detector electronics, the dominant systematic uncer-
tainties on the neutrino-electron correlation coefficient a are a possible charging of the
collimation system, potential Penning discharges in the bottom of the spectrometer, and
a violation of the condition for adiabatic transport, cf. Secs. 5.4.3 and 5.4.4.
209
210 CHAPTER 7. SUMMARY AND OUTLOOK
7.2 Improvements for a Further Beam Time
Thanks to the knowledge of the systematic effects gained in this thesis and the theses of
M. Simson [52], M. Borg [34], and F. Ayala Guardia [53], we are now able to improve the
aSPECT spectrometer to perform a 1% measurement of a in a further beam time at the
ILL (see the next section). The following measures to improve the spectrometer have been
proposed (for details see Chaps. 5 and 6):
• reduce the amplification of the current preamplifier down to about 40% or switch
to, e.g., a logarithmic preamplifier instead, in order to prevent the saturation of the
preamplifier after high-energy electrons, cf. Sec. 5.4.1.
• optimize the pulse shaping, to remove the undershoot of the baseline after high-
energy electrons, cf. Sec. 5.4.1.
• implement a stabilized cooling system for the detector electronics, to improve both
the level and the stability of the electronic noise, cf. Sec. 5.2.1.
• switch to a detector mechanics (see Fig. 3.20a) movable in the x-y-plane, for a
better determination of the position of the proton detector relative to the rest of the
electrode system and the magnetic field, cf. Sec. 5.5.6.
• further improve the vacuum conditions by means of, e.g., an additional external
getter pump, in order to reduce the probability of rest gas ionization and hence also
the amount of uncorrelated background.
• re-open the right side of the decay volume (DV) electrode gr, to increase the pump-
ing speed and simultaneously reduce the influence of work function (WF) inhomo-
geneities on a, cf. Sec. 6.4.2.
• reduce the electrostatic mirror potential, to prevent both a violation of the condition
for adiabatic transport and Penning discharges in the bottom part of the spectrom-
eter, cf. Sec. 5.4.3. Therefore, electrically decouple the wire system (e1) from its
holding electrode (e1b).
• coat the surfaces of the collimation system with, e.g., aluminum or titanium, in order
to prevent a charging of the collimation system, cf. 5.4.4.
• increase the volume of the DV and subsequently the distance between the neutron
beam and the inner surfaces of the DV electrode gr, to reduce the influence of WF
inhomogeneities on a, cf. Sec.6.4.2.
• re-polish and subsequently re-coat the inner surfaces of the analyzing plane (AP)
electrode e14, in order to reduce the influence of WF inhomogeneities on a, cf.
Sec.6.4.1.
• measure the neutron beam profile(s) directly in the DV and improve their quality
by means of the image plate scanner, to reduce the uncertainty in the correction for
the edge effect, cf. Sec. 5.5.6.
• improve the stability of the neutron beam monitor (cf. Sec. 4.2.1), in order to
normalize the measured proton count rates to the measured neutron count rates.
7.3. UPCOMING MEASUREMENTS WITH ASPECT 211
• implement an automated switching of the electrostatic mirror, to, e.g., perform
measurements with a second analyzing plane below and above the AP electrode e14,
cf. Sec. 5.3.2.
• install the nuclear magnetic resonance system developed by F. Ayala Guardia [53],
in order to monitor the stability of the magnetic field ratio rB, of the magnetic fields
in the AP and the DV, during data taking.
• re-examine the influence of non-adiabatic proton motion (by means of MC sim-
ulations), in particular with regard to the height of the main magnetic field, cf.
Sec. 5.5.5.
• additionally integrate the slight energy and angular dependencies (see Fig. 5.47) of
the proton drift into the combined simulations of the edge effect, cf. Sec. 5.5.6.
• increase the simulation statistics (by means of MC simulations on a computer clus-
ter), to reduce the uncertainty in the correction for the edge effect, cf. Sec. 5.5.6.
• develop an ultra-high vacuum compatible scanning Kelvin probe, periodically switch-
ing between DV and AP, in order to monitor the WF values in the DV and the AP
during data taking, cf. Sec. 6.5.2.
• develop a monochromatic ion source to measure in situ the potential difference be-
tween DV and AP, cf. Sec. 3.4.1.
7.3 Upcoming Measurements with aSPECT
Currently, the aSPECT experiment is set up at the PF1B of the ILL. With measurements
in the June/July and the September/October 2011 cycle, we aim to improve the uncer-
tainty in a to 1%, well below the present literature value of 4% [10]. Further emphasis
lies on the study of systematic effects. In particular, we will investigate the remaining
trapped particle background, in order to continue improving the aSPECT spectrometer
to permit a measurement of a with its design accuracy of 0.3%, in a future beam time.
At the same time, we prepare a first measurement of the proton asymmetry parameter
C with the aSPECT spectrometer, derived from the proton recoil spectrum in decays
of polarized neutron. So far, the first and only measurement of C = −0.2377(26) has
been performed with the PERKEO II spectrometer [86, 269]. We aim to improve the
uncertainty in C from currently 1.1% to the 0.1% accuracy level [48].
All these investigations will also help to prepare high-precision measurements of an-
gular correlations in neutron beta decay with the new beam facility PERC1, cf. Sec. 7.4
and App. C.
7.4 The Future with PERC
The new beam station PERC [104], a clean, bright, and versatile source of neutron decay
products, is designed to improve the sensitivity of neutron decay studies by one order
of magnitude. In this way, several symmetry tests based on neutron beta decay data
become competitive [6]. The charged decay products are collected by a strong longitudinal
1Acronym for Proton and Electron Radiation Channel.
212 CHAPTER 7. SUMMARY AND OUTLOOK
Figure 7.1: Scheme of the new facility PERC [270]: Cold neutrons (green) pass through the decay
volume where only a small fraction decays. The decay products (red) are guided by
the strong magnetic field towards the detector (blue). The superconducting coils are
drawn in grey. The equipment for neutron beam preparation, like velocity selector,
polarizer, spin flipper, or chopper, is located in front of the instrument (to the left of
the scheme) and not shown here. For details see [104]. Figure taken from Ref. [7].
magnetic field of 1.5T directly from inside a neutron guide, as can be seen in Fig. 7.1.
This combination provides the highest phase space density of decay products. A magnetic
mirror at 6T serves to perform precise cuts in phase space, reducing related systematic
errors. Systematic errors related to electron spectroscopy have been shown to be on the
level of 10−4, more than 10 times better than that achieved today [104].
PERC is under development by an international collaboration with the Universities of
Heidelberg and Mainz, the Technische Universität München, the Institut Laue-Langevin
in Grenoble, and the Vienna University of Technology. The instrument will be set up at
a new position of the beam facility MEPHISTO of the Forschungs-Neutronenquelle Heinz
Maier-Leibnitz in Munich, Germany.
Depending on the decay parameters studied, the analysis of the decay electrons and
protons will be performed with specialized detectors. For protons, PERC will feed a
charged particle spectrometer, for instance an adapted spectrometer which is based on
our aSPECT spectrometer. The knowledge of the systematic effects gained in this thesis
(see Sec. 5 for details) and the investigations of the patch effect (discussed in Sec. 6) will
help to achieve the aimed uncertainties in the correlation coefficients a and C. For details
see App. C and Ref. [7].
Appendix A
Design of an Anti-Magnetic Screen
This Appendix deals with the design of an anti-magnetic screen for the superconducting
retardation spectrometer aSPECT using COMSOL Multiphysics Electromagnetics Mod-
ule. COMSOL 3.2b was used to model and optimize the anti-magnetic screen to obtain a
design that reduces the exterior magnetic field by a factor of about 10, that does not affect
the internal magnetic field and its homogeneity, and that makes sure that the additional
forces onto the magnet are non-destructive.
The anti-magnetic screen has already been presented in my publication [1]. I am going
to closely follow the description therein and will update the experimental results.
A.1 The Shielding Problem
As explained earlier in Sec. 3.2.2, the spectrometer aSPECT consists of a system of eleven
(cf. Fn. 12 in Chap. 3) superconducting coils placed inside a cylinder with a length of three
meters and a diameter of seventy centimeters (cf. Figs. 3.7 and A.5). The coil system and
its magnetic field are axially symmetric. The magnet generates a strong magnetic field
which varies from 0.6 to 6T along the symmetry axis (see Figs. 3.8 and 4.5), and down
to 5 Gauss in a radial distance of five meters from the decay volume (DV) (see later in
Fig. A.6).
For the latest beam time, the spectrometer aSPECT had to move to the Institut Laue-
Langevin (ILL) in Grenoble, France. So as not to disturb other experiments by its strong
magnetic stray field, an anti-magnetic screen was built that, as a first condition, has to
lower the exterior magnetic field to less than 1 Gauss in a radial distance of 5m, i.e., by
a factor of about 10.
Magnets can be screened by either active or passive shielding. Active shielding refers
to using a second magnet wound in the opposite direction of the main magnet. However,
active shielding works very well only if the second magnet is of substantial larger diameter
than the main magnet. In case of the aSPECT magnet, with coil diameters up to 60 cm
(see Fig. 3.7), an active shield would need too much space.
A passive shield, made of ferromagnetic materials, must have large magnetization. This
creates a substantial magnetic field inside the spectrometer and accordingly additional
forces onto the coils. On the other hand, the aSPECT experiment depends strongly on
the condition that the inhomogeneity of the magnetic field in the DV and in the analyzing
plane (AP) (see Figs. 3.16 and 4.5) does not exceed 10−4. Thus, an additional requirement
of the design is that 2) the disturbance of the magnetic field caused by the shield does not
affect the internal magnetic field and its homogeneity.
213
214 APPENDIX A. DESIGN OF AN ANTI-MAGNETIC SCREEN
The manufacturer of the aSPECT magnet, Cryogenics Ltd., agrees to the anti-
magnetic screen only if the electromagnetic forces onto the coils do not change their sign
and if the relative changes are small. Therefore, a further requirement of the design is
that 3) the additional forces onto the coils are non-destructive in this sense. Obviously,
conditions 1) and 3) are contradictory terms for the strength of the shield.
Figure A.5 shows the spectrometer aSPECT inside of the anti-magnetic screen. Let’s
suppose, the spectrometer is not connected to the antimagnetic screen, and that we could
neglect the gravitational forces, then the magnet would hang in the air only if he is
already placed at a location that nulls the forces onto the coils. Otherwise, if the magnet
is displaced, e.g. just a little bit moved towards a pillar, he will be attracted towards this
pillar. Consequently, the final requirement of the design is that 4) the magnet has to be
adjusted inside the shield.
A.2 Axially Symmetric Shielding
We have begun the design of a passive shield with 2D simulations in axial symmetry.
The magnetic field at any point in space due to a current loop can be obtained using
the law of Biot Savart, integrated over a circular current loop [170], i.e. by numerical
integration of analytical formulas. Consequently, we proved first that our semi-analytical
calculations of the magnetic field and a shield of linear permeability are in agreement with
COMSOL. For this comparison, the spectrometer aSPECT was enclosed by a cylinder
with a length of 400 cm and a diameter of 190 cm with a lid on top and on bottom, both
with a thickness of 10 cm, what corresponds to a mass of 20 tons.
Ferromagnetic materials have non-linear permeability, described by a so called B-H-
curve. The shape of this curve, especially the initial permeability, the maximal perme-
ability and the saturation, are strongly correlated with the special kind of the magnetic
material (construction steel, transformer steel, mumetal, etc.), and depend on the thickness
and the pretreatment of the material. Thus, the 2D modeling was completed by simu-
lations of different materials of non-linear permeability. In these simulations the shield
factor (defined as the ratio of the magnetic fields in radial direction without and with
shield) reached a constant value from a radial distance of 5m on, as expected for physical
reasons. Figure A.1 shows a simulation for an axially symmetric model in COMSOL. In
this design the spectrometer aSPECT is enclosed by a cylinder as described above, made
of the special steel grade RTM3.
Although such designs would provide big enough shield factor (up to about 60 for
RTM3), they make the support of the spectrometer very difficult. On the other hand,
for a box instead of the cylinder we expect a lower but still sufficient large shield factor.
Thus, availability and price of ferromagnetic materials suggest to using a box instead of
the cylinder, i.e. a non-axially symmetric design what required a 3D simulation to be done
(see the following section).
As mentioned above, a shield made of ferromagnetic materials creates additional forces
onto the coils. Hence, the requirements 3) and 4) demand a minimum size on a passive
shield, what required electromagnetic force calculations to be done simultaneously (for
details see Sec. A.4).
A.3. NON-AXIALLY SYMMETRIC SHIELDING 215
Figure A.1: The magnetic field for a 2D shield made of RTM3. The boundary of the shield is
drawn in yellow. Figure taken from Ref. [1].
A.3 Non-axially Symmetric Shielding
We continued the design of a passive shield with 3D simulations, in the application modes
that use either the magnetic vector potential A or the magnetic scalar potential Vm.
The conditions 1) – 4) require a fine mesh in radial direction (at least up to 10m),
close to the z-axis, in the DV, in the AP and inside the coils, at the same time. This
requirement is equivalent to a fine mesh in the whole space and so nearly impossible. In
order to refine the mesh in our regions of interest we simulated only an eighth of the
216 APPENDIX A. DESIGN OF AN ANTI-MAGNETIC SCREEN
Figure A.2: The final design for passive shielding. A combination of both construction steel
(plates) and ARMCO iron (pillar) provides a sufficient reduction of the exterior mag-
netic field without considerable effect on the internal magnetic field and its homo-
geneity, as can be seen from Fig. A.4. Figure taken from Ref. [1].
geometry taking advantage of the symmetry of the shield (see Fig. A.2 and also A.5).
That way, the radial direction turns into two boundaries and the z-axis into an edge of
the geometry, and the volume of the coils is reduced considerably. Furthermore, for this
purpose we have to use “magnetic insulation” n × A = 0 as boundary condition on the
symmetry planes.
The 3D simulations of axially symmetric shields were in agreement with the corre-
sponding 2D modeling results, as well. Thus, we have finished the design with 3D simula-
tions of non-linear materials. We modeled a series of shields with the following geometry
and variable parameters: A frame with 2 plates of variable cross-section and thickness,
A.3. NON-AXIALLY SYMMETRIC SHIELDING 217
Figure A.3: The magnetic field in the symmetry plane that cuts through a pillar of the anti-
magnetic screen for the model in Fig. A.2. Figure taken from Ref. [1].
with a hole of variable diameter at the center, and n · 4 pillars between the plates, in
their corners, each quartet of variable cross-section and length. Figures A.2 and A.3 show
a simulation for a non-axially symmetric model in COMSOL. In this final design the
magnet is enclosed by a combination of construction steel S235JRG2 (2 plates each of
volume 180× 180× 10 cm2 and with a hole of a diameter of 50 cm) and ARMCO iron (4
pillars each of volume 20 × 20 × 400 cm2), what corresponds to a mass of 10 tons. This
combination of different materials was chosen only because of availability of materials. The
subdomain plot in Fig. A.3 illustrates already the saturation effect (BARMCOsat = 2.13T) in
highmagnetic field environments, what explains the impracticalness of high-permeability
materials like transformer steel or mumetal in case of the spectrometer aSPECT, also.
In summary, this design meets the requirement to reduce the exterior magnetic field by a
218 APPENDIX A. DESIGN OF AN ANTI-MAGNETIC SCREEN
(a) (b)
Figure A.4: Expected influence on the internal magnetic field: The magnetic field on the z-axis
(a) in the DV and (b) in the AP for the model in Fig. A.2, without (red line) and
with anti-magnetic screen (blue line).
shield factor of 8.2 from a radial distance of 10m on, and can be compared to a shield factor
of about 30 for a cylinder as described in Sec. A.2 made of ARMCO iron. Moreover, the
influence on the internal magnetic field is quite small, i.e. the shape and the homogeneity
of the magnetic field in the DV and in the AP remain unaffected, only the ratio of the
magnetic fields in these regions changes (cf. Fig. A.4).
A.4 Electromagnetic Force Calculation
The 3D simulations of a passive shield were accompanied by electromagnetic force calcu-
lations, in the application mode that uses the magnetic scalar potential Vm.
The additional force onto a coil in the external magnetic field Bext caused by the shield
is result from s∫ubdomain integration to
FV = dV J×Bext , (A.1)
V
where V denotes the volume of the coil and J its current density [17
Table A.1 shows a simulation of the electromagnetic forces on√0].to an eighth of the
aSPECT coils (denoted by c1 – c11 as in Fig. 3.7), where F = F 2 + F 2r x y and ϕ =
arcsinFy/Fr. As one can see from the total sum of additional forces ∆Fz, the magnet is
already placed at a location that nulls the additional forces caused by the shield. Moreover,
none of the forces changes its sign and the relative changes are quite small, both as
desired (except for the already small radial forces onto the coils c2c and c4c). Besides, by
displacement of the anti-magnetic screen relative to the aSPECT magnet, we calculated
the gradient of the additional forces to about 300N per 1 cm displacement of the magnet
along the z-axis.
Without shield, the total sum of forces Fz onto the coils has to be zero, in contradiction
to a calculated sum of 1 kN, cf. Table A.1. The angle ϕ of the radial forces is 22.5°, as
A.5. EXPERIMENTAL RESULTS 219
Table A.1: The electromagnetic forces onto an eighth of the coils without shield (Fr and Fz w/o
shield) and the additional forces caused by the anti-magnetic screen (∆Fr and ∆Fz)
for the model in Fig. A.2. Data taken from Ref. [1].
coil Fr (kN) ϕ (°) Fz (kN) ∆Fr (kN) ∆Fz (kN)
w/o shield w/o shield
c1 65 22.5 15 0.5 -0.02
c2a 143 22.5 17 0.5 -0.03
c2b 61 22.5 10 0.3 -0.02
c2c 6 22.5 53 -1.4 -0.10
c4a 100 22.5 -10 0.3 -0.02
c4b 73 22.5 -11 0.3 -0.02
c4c 0.3 16.8 -44 1.2 -0.09
c6 44 22.5 -4 0.2 -0.01
c7 66 22.5 2 0.4 -0.02
c8 109 22.5 -26 0.7 -0.05
c9 16 22.5 0.03 0.3 0
c10 379 22.4 104 1.6 0.20
∑c11 256 22.5 -105 1.6 0.20
1 0
expected (except for c4c, where small errors of the small components Fx and Fy can
explain a deviation of about 6°). However, errors of that order are tolerable.
A.5 Experimental Results
On the basis of the electromagnetic force calculations the spectrometer aSPECT was
centered inside the anti-magnetic screen (see Fig. A.5). For that purpose, the magnet
was hanged on a crane and separated from its suspension to the anti-magnetic screen
meanwhile the magnetic field was ramped up very slowly. The obtained unstable position
of equilibrium deviates from the calculations by only 1 cm along the z-axis, a deviation
that corresponds to a small force of 300N.
Moreover, we verified the simulations by measuring the reduction of the exterior mag-
netic field by a factor of about 7 (in a radial distance of 5.6m from the DV), as can be
seen in Fig. A.6.
The last step was to prove the simulations by measuring the internal magnetic field
on and close to the z-axis with our Hall probe (cf. Fn. 7 in Chap. 4). Figure A.7 shows
a measurement along the z-axis of the spectrometer. In the AP, the experimental data
points perfectly match the expected values from our simulations. In the DV, by contrast,
the experimental data points exceed the expected values from our simulations. However,
both the shape and the homogeneity of the magnetic field in the DV remain unaffected,
while only the ratio of the magnetic fields in the DV and the AP changes. We believe that
the minor deviation can be attributed to the moderate knowledge of the B-H-curves.
220 APPENDIX A. DESIGN OF AN ANTI-MAGNETIC SCREEN
Figure A.5: Setup of the spectrometer aSPECT inside the anti-magnetic screen, a combination
of construction steel (plates of cross-section 180× 180× 10 cm2 and thickness 10 cm)
and ARMCO iron (pillars of cross-section 20× 20× 10 cm2 and length 400 cm). The
shield provides a sufficient reduction of the exterior magnetic field (cf. Fig. A.6).
Photograph taken from Ref. [1].
A.5. EXPERIMENTAL RESULTS 221
Figure A.6: Influence on the exterior magnetic field: The magnetic field in radial direction for the
model in Fig. A.2, without (red) and with anti-magnetic screen (blue line). The green
stars indicate a measurement between two pillars of the anti-magnetic screen.
(a) (b)
Figure A.7: Influence on the internal magnetic field: The magnetic field on the z-axis (a) in the
DV and (b) in the AP for the model in Fig. A.2, without (red line) and with anti-
magnetic screen (blue line). The green stars indicate a measurement along the z-axis
of the spectrometer.
222 APPENDIX A. DESIGN OF AN ANTI-MAGNETIC SCREEN
Appendix B
Details of the Kelvin Probe Samples
Table B.1 presents a summary of the samples studied to date, to be scanned, or in prepa-
ration, including details of the substrates, surface treatments, and coatings. The first
column gives the name of the sample, i.e., the common abbreviation for the substrate plus
a serial number. The second column specifies the substrate type. The third and fourth
column indicate if the substrate was subjected to welding or brazing in the mechanical
workshop (MW) at the University of Mainz. The fifth and sixth column list the method of
polishing along with the respective manufacturer. The seventh to eleventh column specify
the coating, listed from substrate surface to outer surface, along with the respective man-
ufacturer. The last column gives a short comment, if appropriate. All abbreviations are
standard nomenclature for chemical elements, except OFHC Cu (oxygen-free high ther-
mal conductivity copper), WB (white bronze), and IPA (isopropyl alcohol). In addition,
the following abbreviations are used: PC (polishing compound), BW (buffing wheel), Sp
(Physical Vapor Deposition, i.e., sputtering), EP (electroplating, i.e., galvanically), and
EC (electroplating, i.e., electro-chemical). The remaining abbreviations stand for the
various manufacturers, i.e., MVG (Metallveredelungs GmbH in Wiesbaden, Germany),
AG (Adler Galvano GmbH in Mainz, Germany), GG (Galvanotechnik Gierich GmbH in
Neuss, Germany), SMV (Sigrist Metallveredelung GmbH in Pforzheim, Germany), TL
(Dr.Thorsten Lauer at the University of Mainz), and ECG (EC EuropCoating GmbH in
Hohenlockstedt, Germany).
223
224 APPENDIX B. DETAILS OF THE KELVIN PROBE SAMPLES
Table B.1: Details of the Kelvin Probe samples studied, to be scanned, or in preparation.
Ref.no. Substrate Welding Polishing Coat. 1st layer 2nd layer Comment
method Co. method Co. techn. [µm] Co. [µm] Co.
Cu1 OFHC Cu PC green rouge MVG Spa Ag 2 TL 24C Au 0.2 TL cleaned w/ IPA
Cu2 OFHC Cu PC green rouge MVG Spa Ag 2 TL 24C Au 0.2 TL
Cu3 OFHC Cu nylon fleece MW EP Ag 10 AG 24C Au 1 AG
Cu4 OFHC Cu nylon fleece MW EP Ag 10 AG 24C Au 1 AG
OFHC Cu,
Cu5 OFHC Cu cont. seam MW nylon fleece MW EP Ag 10 AG 24C Au 1 AG cannot be scanned
OFHC Cu,
Cu6 OFHC Cu cont. seam MW nylon fleece MW EP Ag 10 AG 24C Au 1 AG cannot be scanned
Cu7 OFHC Cu PC talcum AG Spa Ag 2 TL 24C Au 0.2 TL
Cu8 OFHC Cu PC talcum AG Spa Ag 2 TL 24C Au 0.2 TL cleaned w/ IPA
Cu9 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 2 AG
Cu10 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 2 AG
Cu11 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 1 AG
Cu12 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 1 AG
Cu13 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 5 AG
Cu14 OFHC Cu PC talcum AG EP Ag 10 AG 24C Au 5 AG
Cu15 OFHC Cu PC blue, BW GG EP Ag 10 GG 24C Au 2 GG 0.3% Co
Cu16 OFHC Cu PC blue, BW GG EP Ag 10 GG 24C Au 2 GG 0.3% Co
Cu17 OFHC Cu PC blue, BW GG EP Ag 10 GG 24C Au 2 GG 0.3% Co
Cu18 OFHC Cu PC blue, BW GG EP WB 1 GG Pt 1 GG
Cu19 OFHC Cu PC blue, BW GG EP WB 1 GG Pt 1 GG
Cu20 OFHC Cu PC blue, BW GG EP WB 1 GG Pt 1 GG
aOnly one side of the substrate was sputter coated.
225
Ref.no. Substrate Brazing Polishing Coat. 1st layer 2nd layer Comment
method Co. method Co. techn. [µm] Co. [µm] Co.
Cu21 OFHC Cu PC white, Inlett SMV Spa Ag 2 TL 24C Au 0.2 TL cleaned w/ IPA
Cu22 OFHC Cu PC white, Inlett SMV Spa Ag 2 TL 24C Au 0.2 TL
Cu23 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV Rh 0.2 SMV
Cu24 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV Rh 0.2 SMV to be scanned
Cu25 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV Pt 0.2 SMV
Cu26 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV Pt 0.2 SMV
Cu27 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV 24C Au 2 SMV
Cu28 OFHC Cu PC white, Inlett SMV EP Ag 10 SMV 24C Au 2 SMV
EC Ni 1 AG 1st layer electro-
Cu29 OFHC Cu manually AG EP 24C Au 2 AG chemical, magnetic
Cu30 OFHC Cu manually AG EP Ni 1 AG 24C Au 2 AG magnetic
Cu31 OFHC Cu manually AG EP Ni 10 AG 24C Au 2 AG magnetic
Cd-free Ag, nylon fleece MW
Cu32 OFHC Cu cont. seam MW PC talcum AG EP Ag 10 AG 24C Au 1 AG
Cd-free Ag, nylon fleece MW
Cu33 OFHC Cu cont. seam MW PC talcum AG EP Ag 10 AG 24C Au 1 AG
Cd-free Ag, nylon fleece MW
Cu34 OFHC Cu cont. seam MW PC talcum AG
Cd-free Ag,
Cu35 OFHC Cu cont. seam MW PC talcum AG EP Ag 10 AG 24C Au 1 AG to be scanned
Cd-free Ag,
Cu36 OFHC Cu cont. seam MW PC talcum AG EP Ag 10 AG 24C Au 1 AG to be scanned
226 APPENDIX B. DETAILS OF THE KELVIN PROBE SAMPLES
Ref.no. Substrate Brazing Polishing Coat. 1st layer 2nd layer Comment
method Co. method Co. techn. [µm] Co. [µm] Co.
Cd-free Ag, 2
Cu37 OFHC Cu capillary joints MW PC talcum AG EP Ag 10 AG 24C Au 1 AG to be scanned
Cd-free Ag, 2
Cu38 OFHC Cu capillary joints MW PC talcum AG EP Ag 10 AG 24C Au 1 AG to be scanned
Cu39 OFHC Cu PC AG Sp TiN 0.8 ECG 1st layer failed
1st layer failed,
Cu40 OFHC Cu PC AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
1st layer failed,
Cu41 OFHC Cu PC AG Sp TiN 0.8 ECG Ni 1 AG 2nd layer non-adhere
1st layer failed,
Cu42 OFHC Cu PC AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
2% Ni, magnetic,
Cu43 OFHC Cu PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
2% Ni, magnetic,
Cu44 OFHC Cu PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
2% Ni, magnetic,
Cu45 OFHC Cu PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
2% Ni, magnetic,
Cu46 OFHC Cu PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
Cu47 OFHC Cu PC AG Sp Ag 10 TL 24C Au 1 TL coating projected
Cu48 OFHC Cu PC AG Sp Ag 10 TL 24C Au 1 TL coating projected
Cu49 OFHC Cu PC AG Sp Ag 2 TL 24C Au 0.2 TL coating projected
Cu50 OFHC Cu PC AG Sp Ag 2 TL 24C Au 0.2 TL coating projected
227
Ref.no. Substrate Brazing Polishing Coat. 1st layer 2nd layer Comment
method Co. method Co. techn. [µm] Co. [µm] Co.
Cu51 OFHC Cu PC AG Sp Cr 2 TL 24C Au 0.2 TL coating projected
Cu52 OFHC Cu PC AG Sp Cr 2 TL 24C Au 0.2 TL coating projected
Cu53 OFHC Cu manually SMV EP Ag 10 SMV Pt 0.2 SMV
Cu54 OFHC Cu manually SMV EP Ag 10 SMV Pt 0.2 SMV
Cu55 OFHC Cu manually SMV EP Ag 10 SMV 24C Au 1 SMV to be scanned
Cu56 OFHC Cu manually SMV EP Ag 10 SMV 24C Au 1 SMV to be scanned
Cu57 OFHC Cu manually SMV EP Ag 10 SMV 24C Au 2 SMV to be scanned
Cu58 OFHC Cu manually SMV EP Ag 10 SMV 24C Au 2 SMV to be scanned
Cu59 OFHC Cu PC AG Sp TiW 2 TL 24C Au 0.2 TL coating projected
Cu60 OFHC Cu PC AG Sp TiW 2 TL 24C Au 0.2 TL coating projected
Ti1 Ti grade 1 Spa 24C Au 0.2 TL
Ti2 Ti grade 1 Spa 24C Au 0.2 TL cleaned w/ IPA
Ti3 Ti grade 1 PC talcum AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
Ti4 Ti grade 1 PC talcum AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
Ti5 Ti grade 1 PC talcum AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
Ti6 Ti grade 1 PC talcum AG Sp TiN 0.8 ECG 24C Au 0.2 TL 2nd layer projected
Ti7 Ti grade 1 PC AG Sp 24C Au 0.2 TL coating projected
Ti8 Ti grade 1 PC AG Sp 24C Au 0.2 TL coating projected
2% Ni, magnetic,
Ti9 Ti grade 1 PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
2% Ni, magnetic,
Ti10 Ti grade 1 PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
228 APPENDIX B. DETAILS OF THE KELVIN PROBE SAMPLES
Ref.no. Substrate Brazing Polishing Coat. 1st layer 2nd layer Comment
method Co. method Co. techn. [µm] Co. [µm] Co.
2% Ni, magnetic,
Ti11 Ti grade 1 PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
2% Ni, magnetic,
Ti12 Ti grade 1 PC AG Sp TiN 0.8 ECG 23.5C Au 0.1 ECG to be scanned
Glass1 glass Spa Ag 0.2 TL
Glass2 glass Spa Au 0.2 TL
Glass3 glass Sp Au 0.2 TL coating projected
Glass4 glass Sp Au 0.2 TL coating projected
Appendix C
The New Facility PERC
The Proton and Electron Radiation Channel (PERC) is a new type of beam station for
the measurement of angular correlations in the beta decay of free neutrons. In contrast to
existing neutron decay spectrometers, PERC is a user instrument which delivers at its exit
not neutrons but an intense beam of decay electrons and protons, under well defined and
precisely variable conditions. Depending on the observable to be investigated, different
secondary spectrometers can be used, for instance an adapted proton spectrometer which
is based on our aSPECT spectrometer. Thus, with PERC, we can measure the shapes and
magnitudes of electron or proton energy spectra from polarized or unpolarized neutron
decay, in pulsed or continuous neutron beam mode, with or without electron spin analysis.
Many quantities can be derived from such spectra, some for the first time [7, 104].
After a short description of its measurement principles, we will present the uncertain-
ties and systematics of PERC. We note that the achievable accuracies of the correlation
coefficients a and C depend heavily on the systematic uncertainties in the spectroscopy of
decay protons, which have not yet been completely analyzed, cf. Sec. C.3.
PERC has already been presented in my publication [7]. I am going to closely follow
the description therein and will update some details on proton spectroscopy.
C.1 Measurement Principles and Instrument
PERC is a beam station, which delivers at its exit neutron decay products. The set-up is
schematically shown in Fig. 7.1 and in Refs. [7, 104]; its design principles are thoroughly
discussed in Ref. [104]. Here, we will only summarize the essential parts: Cold neutrons
pass through an 8 meters long neutron guide, with a cross section of 6×6 cm2, where about
106 neutrons decay per second and per meter of guide. The neutron guide is surrounded
by a superconducting solenoid (B0 = 1.5T) of equal length. Decay electrons and protons
are guided by the strong longitudinal magnetic field towards the electron/proton (e/p)
detection system, i.e., detectors specialized for certain tasks. This combination provides
the highest phase space density of decay products. At the end of the neutron guide, the
decay products can be separated from the neutron beam by means of bending coils. In
the e/p selector, the decay products pass a region of strongly enhanced magnetic field
B1 > B0 (B1 = 3− 6T) before they reach the detector (at, e.g., B2 = 0.5T). The field B1
acts as√a magnetic mirror and transmits only the fraction (1− cos θC) /2 (≈ B0/4B1, forB0 < B1) of all decay products, namely those emitted upstream under angles θ0 ≤ θC =
arcsin B0/B1 to the z−axis. Field variations B(z) must be slow enough such that the
229
230 APPENDIX C. THE NEW FACILITY PERC
Figure C.1: The black and red line indicate the transmission function of PERC for protons in
unpolarized neutron decay (preliminary), for B1 = 6T and aSPECT as detection
system, with B2 = BA = 0.4T and UA = 50V respectively UA = 400V. For eluci-
dation, the green line shows the proton recoil spectrum with the recommended value
for a = −0.103 [10].
decay electrons and protons are transported adiabatically. Then, the magnetic transport
no longer depends on the energy of the decay products.
C.2 Measurement Uncertainties and Systematics
The magnetic mirror serves to limit the phase space precisely, reducing related systematic
errors. Systematic errors related to electron spectroscopy have been shown to be on the
level of 10−4, more than 10 times better than that achieved today [104]. There is one
exception related to the knowledge of the neutron beam polarization, where presently the
error is on the 10−3 level [44]. Techniques for the polarization of a roughly monochromatic
(∆λ/λ ≈ 10%) cold neutron beam will be improved towards the 10−4 level.
Further details on the sensitivity and the applications of PERC may be found in [104].
For details on the neutron beam preparation and the polarization analysis we refer to
[271–273], respectively.
C.3 Dominant Uncertainties in the Analysis of Protons
Depending on the decay parameters studied with PERC, the analysis of the decay electrons
and protons will be performed with specialized detectors. As far as electrons are concerned,
this can be done with an energy sensitive detector. For protons, PERC will feed a charged
particle spectrometer, for instance an adapted spectrometer which can partially be based
on the aSPECT detection system. The most important associated systematic uncertainties
for aSPECT as detection system for PERC are:
Homogeneity of the magnetic field: In the adiabatic approximation, the transmis-
sion function as shown in Fig. C.1 can be calculated analytically:
C.3. DOMINANT UNCERTAINTIES IN THE ANALYSIS OF PROTONS 231
 0 √ ( ) , if T0 ≤ eUA
FPERC
1
tr (T0;UA) =  1−√1−
B0 1− eUAB T , if otherwise , (C.1)2 2 0
1− 1− B0B , if T
eUA
0 ≥
1 1−B2/B1
where UA is the barrier voltage of aSPECT. Our initial calculations indicate that the
magnetic fields B0, B1, and B2 = BA must be controlled at the level of ∆B0/B0 <
1%, ∆B −4 −41/B1 = 1 × 10 , and ∆B2/B2 = 1 × 10 , respectively, in order to keep
∆a/a < 0.1%. The condition ∆B0/B0 < 1% can be fullfilled in pulsed neutron
beam mode. Then, the decay products will be counted in the detector only while the
neutron pulse is fully contained within the central, homogeneous part of the magnetic
field. The magnetic field calculations show that the condition ∆B1/B1 = 1 × 10−4
is fulfilled within the flux of neutron decay products [7]. For aSPECT, it has been
shown that ∆B2/B2 is on the level of 10−4 [53].
Homogeneity of the electric field: Our first estimates demonstrate that the electric
potential between the decay volume, the e/p selector, and the electrostatic barrier
will have to be known with an accuracy of better than 10mV, comparable to aSPECT
and Nab [99, 163].
Doppler effect due to neutron motion: Unlike in aSPECT or Nab, the Doppler ef-
fect is not negligible as the neutron beam is collinear to the detection system. For a
Gaussian neutron spectrum, preliminary estimates show that the mean neutron en-
ergy has to be known with a precision of better than 10−2, whereas the uncertainty
in width of the spectrum is negligible [274]. An approximately Gaussian neutron
spectrum can be obtained, e.g., by a neutron velocity selector.
Adiabaticity of the proton motion: For the determination of the neutrino-electron
correlation coefficient a, adiabatic transport must be guaranteed between the decay
volume and the electrostatic barrier. Equation (3.7) is a first estimate for the adia-
baticity of the proton motion. The magnetic field calculations show that adiabatic
transport is guaranteed along the particle trajectories [7]. We note that electric
fields are not considered in this ansatz.
Residual gas: Simulations for aSPECT and Nab [163] have demonstrated that changes
in the extracted value of a are tolerable if the residual gas pressure can be reduced
to 10−8 mbar independent of the type of gas, and negligible if the pressure can be
reduced to 10−9 mbar.
Particle trapping: Surface potential variations in the order of 100mV have been found
in electrodes prepared within the aSPECT project, cf. Ref. [33] and Sec. 6. Such
variations can lead to local field extrema within the decay volume, the e/p selector,
or the electrostatic barrier, and can therefore give rise to potential penning traps.
For the same reason, local magnetic field minima must be avoided.
All these systematic effects are on the simulation agenda, and will be analyzed in due
course. Measures to reduce and to precisely control systematic errors, like an online nuclear
magnetic resonance system for monitoring of the magnetic field with 10−4 accuracy [53]
or investigations of the patch effect with surface potential variations < 10mV (see Sec. 6
and Ref. [243]), have already been initiated.
232 APPENDIX C. THE NEW FACILITY PERC
Bibliography
[1] G. Konrad et al., Design of an Anti-Magnetic Screen for the Neutron Decay
Spectrometer aSPECT, in Proceedings of the European Comsol Conference, Greno-
ble, 2007, pp. 241–245, 2007, http://www.comsol.com/papers/3276/. (Cited on
pages vii, 64, 213, 215, 216, 217, 219, and 220.)
[2] Report for the TRAPSPEC JRA-11 in EURONS on task T-J11-1 (Simulations and
calculations), http://www.gsi.de/documents/DOC-2009-Mar-290-19.pdf, 2008.
(Cited on page vii.)
[3] Report for the TRAPSPEC JRA-11 in EURONS on task T-J11-8 (Neutron decay re-
tardation spectrometer), http://www.gsi.de/documents/DOC-2009-Mar-290-26.
pdf, 2008. (Cited on pages vii and 72.)
[4] Measurement of the Proton Spectrum in free Neutron Decay, ILL Experimental
Report, 2009. (Cited on page vii.)
[5] G. Konrad et al., Nucl. Phys. A 827, 529c (2009). (Cited on pages vii, 4, and 197.)
[6] G. Konrad, W. Heil, S. Baeßler, D. Počanić, and F. Glück, Impact of
Neutron Decay Experiments on Non-Standard Model Physics, in Proceedings of
the 5th International Conference Beyond 2010, Cape Town, South Africa, edited by
H. V. Klapdor-Kleingrothaus, I. V. Krivosheina, and R. Viollier, pp.
660–672, World Scientific, Singapore, 2011, arXiv:1007.3027v2 [nucl-ex]. (Cited on
pages vii, 2, 5, 28, and 211.)
[7] G. Konrad et al., J. Phys.: Conf. Series (2011), accepted. (Cited on pages vii,
212, 229, and 231.)
[8] J. Chadwick, Nature 129, 312 (1932). (Cited on page 1.)
[9] Nobel Lectures, Physics 1922-1941, Elsevier Publishing Company, Amsterdam, 1965.
(Cited on page 1.)
[10] Nakamura K. et al. (Particle Data Group), J. Phys. G: Nucl. Part. Phys. 37,
075021 (2010), and 2011 partial update for the 2012 edition (http://pdg.lbl.go).
(Cited on pages 1, 2, 3, 4, 6, 8, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 29, 34, 39,
40, 49, 50, 51, 52, 71, 79, 81, 101, 103, 104, 106, 107, 108, 113, 117, 126, 127, 128,
129, 137, 139, 145, 146, 147, 150, 153, 154, 155, 157, 201, 204, 207, 211, and 230.)
[11] S. L. Glashow, Nucl. Phys. 22, 579 (1961). (Cited on page 1.)
[12] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964). (Cited on pages 1 and 6.)
233
234 BIBLIOGRAPHY
[13] A. Salam and J. C. Ward, Physics Letters 13, 168 (1964). (Cited on page 1.)
[14] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). (Cited on page 1.)
[15] A. Salam, Elementary Particle Physics: Relativistic Groups and Analyticity, Eighth
Nobel Symposium, Stockholm, p. 367, Almquvist and Wiksell, 1968. (Cited on
page 1.)
[16] F. J. Hasert et al., Phys. Lett. B 46, 138 (1973). (Cited on page 1.)
[17] N. Severijns, M. Beck, and O. Naviliat-Čunčić, Rev. Mod. Phys. 78, 991
(2006). (Cited on pages 2, 5, 20, 28, 29, and 33.)
[18] H. Abele, Prog. Part. Nucl. Phys. 60, 1 (2008). (Cited on pages 2, 3, 4, 5, and 23.)
[19] D. Dubbers and M. G. Schmidt, Rev. Mod. Phys. (2011), in print. (Cited on
pages 2, 3, 5, 13, 23, and 26.)
[20] P. Herczeg, Prog. Part. Nucl. Phys. 46, 413 (2001). (Cited on pages 2 and 5.)
[21] M. J. Ramsey-Musolf and S. Su, Phys. Rev. 456, 1 (2008). (Cited on page 2.)
[22] J. S. Nico, J. Phys. G: Nucl. Part. Phys. 36, 104001 (2009). (Cited on pages 2, 3,
5, 23, and 40.)
[23] T. Soldner et al., editors, Proceedings of the International Workshop on Particle
Physics with Slow Neutrons, Grenoble, France, 2008, Nucl. Instr. and Meth. A 611,
2009. (Cited on pages 2 and 23.)
[24] J. D. Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957).
(Cited on pages 2, 10, and 11.)
[25] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). (Cited on pages 2 and 8.)
[26] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). (Cited on
pages 2 and 8.)
[27] F. Glück, J. Joó, and J. Last, Nucl. Phys. A 593, 125 (1995). (Cited on pages 2,
9, 10, 11, 12, 13, 15, 20, 21, and 31.)
[28] F. Glück, Phys. Lett. B 376, 25 (1996). (Cited on pages 2 and 12.)
[29] J. Liu et al., Phys. Rev. Lett. 105, 181803 (2010),
see also PRL 105 219903 (errat). (Cited on pages 3, 4, and 23.)
[30] H. Abele, Nucl. Instr. and Meth. A 611, 193 (2009). (Cited on pages 3, 23, and 28.)
[31] D. Mund, Messung der Betaasymmetrie A im Neutronenzerfall, PhD thesis,
Ruprecht-Karls-Universität Heidelberg, 2006. (Cited on pages 3 and 23.)
[32] H. Mest, Measurement of the β-Asymmetry in the Decay of Free Polarized Neu-
trons with the Spectrometer PERKEO III, PhD thesis, Ruperto-Carola University
of Heidelberg, 2011. (Cited on pages 3, 4, 21, 23, and 131.)
BIBLIOGRAPHY 235
[33] S. Baeßler et al., Eur. Phys. J. A 38, 17 (2008). (Cited on pages 3, 4, 38, 56,
57, 64, 75, 81, 106, 107, 108, 197, 209, and 231.)
[34] M. Borg, The Electron Antineutrino Angular Correlation Coefficient a in Free
Neutron Decay: Testing the Standard Model with the aSPECT-Spectrometer, PhD
thesis, Johannes Gutenberg-Universität Mainz, 2011. (Cited on pages 3, 4, 53, 57,
72, 75, 81, 85, 87, 92, 95, 96, 101, 105, 106, 107, 109, 110, 114, 116, 119, 120, 123,
129, 135, 140, 141, 149, 152, 165, 166, 209, and 210.)
[35] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 96, 032002 (2006). (Cited on
pages 3 and 13.)
[36] A. Czarnecki, W. J. Marciano, and A. Sirlin, Phys. Rev. D 70, 093006 (2004).
(Cited on pages 3, 13, and 25.)
[37] P. Bopp et al., Phys. Rev. Lett. 56, 919 (1986). (Cited on pages 3 and 4.)
[38] B. G. Erozolimsky et al., Phys. Lett. B 412, 240 (1997).
[39] P. Liaud et al., Nucl. Phys. A 612, 53 (1997). (Cited on pages 3 and 4.)
[40] C. Stratowa, R. Dobrozemsky, and P. Weinzierl, Phys. Rev. D 18, 3970
(1978). (Cited on pages 3, 4, 36, 37, 81, and 209.)
[41] J. Byrne et al., J. Phys. G: Nucl. Part. Phys. 28, 1325 (2002). (Cited on pages 3,
4, 37, 43, 72, 73, 81, and 209.)
[42] H. Abele et al., Phys. Lett. B 407, 212 (1997). (Cited on page 4.)
[43] H. Abele et al., Phys. Rev. Lett. 88, 211801 (2002). (Cited on pages 4 and 27.)
[44] M. Kreuz et al., Nucl. Instr. and Meth. A 547, 583 (2005). (Cited on pages 3
and 230.)
[45] S. Gardner and C. Zhang, Phys. Rev. Lett. 86, 5666 (2001). (Cited on page 3.)
[46] O. Zimmer et al., Nucl. Instr. and Meth. A 440, 548 (2000). (Cited on pages 3,
38, 43, 50, 51, 52, 70, 92, 93, 197, and 209.)
[47] F. Glück et al., Eur. Phys. J. A 23, 135 (2005). (Cited on pages 3, 4, 38, 43, 49,
53, 57, 61, 62, 70, 72, 73, 74, 75, 76, 78, 79, 93, 114, 149, 151, 165, 197, and 209.)
[48] O. Zimmer, private communication, 2010,
see also: http://gepris.dfg.de/gepris/OCTOPUS/;jsessionid=
5CEA2AFD3040D036E9433144DAEEAF54?module=gepris&task=
showDetail&context=projekt&id=167643526. (Cited on pages 3, 4, 24, and 211.)
[49] R. Muñoz Horta, First measurements of the aSPECT spectrometer., PhD thesis,
Johannes Gutenberg-Universität Mainz, 2011. (Cited on pages 4, 26, 38, 56, 64, 81,
106, and 107.)
[50] M. Simson et al., Nucl. Instr. and Meth. A 611, 203 (2009), in Proceedings of the
International Workshop on Particle Physics with Slow Neutrons, Grenoble, 2008,
arXiv:0811.3851v1. (Cited on pages 4, 57, and 197.)
236 BIBLIOGRAPHY
[51] M. Simson et al., Nucl. Instr. and Meth. A 581, 772 (2007). (Cited on pages 4, 64,
65, 66, and 91.)
[52] M. Simson, Measurement of the electron antineutrino angular correlation coefficient
a with the neutron decay spectrometer aSPECT, PhD thesis, Technische Universität
München, 2010. (Cited on pages 4, 43, 53, 64, 65, 66, 68, 70, 75, 79, 80, 81, 85, 87,
90, 91, 92, 94, 95, 96, 98, 99, 101, 105, 106, 109, 110, 114, 115, 116, 118, 119, 120,
121, 122, 123, 124, 125, 129, 159, 160, 166, 209, and 210.)
[53] F. Ayala Guardia, Calibration of the retardation spectrometer aSPECT, PhD
thesis, Johannes Gutenberg-Universität Mainz, 2011, in progress. (Cited on pages 4,
43, 53, 62, 71, 72, 81, 87, 88, 89, 90, 101, 148, 149, 165, 210, 211, and 231.)
[54] J. Byrne, Neutrons, Nuclei and Matter: An Exploration of the Physics of Slow
Neutrons, Institute of Physics Publ., Bristol, 1995. (Cited on page 5.)
[55] F. Halzen and A. D. Martin, Quarks and Leptons: An Introductory Course in
Modern Particle Physics, volume 1, Wiley, 1st edition, 1984. (Cited on page 5.)
[56] P. Renton, Electroweak interactions: an introduction to the physics of quarks and
leptons, Cambridge University Press, Cambridge, 1990.
[57] W. N. Cottingham and D. A. Greenwood, An Introduction to the Standard
Model of Particle Physics, Cambridge University Press, Cambridge, 2nd edition,
2007. (Cited on pages 5 and 8.)
[58] V. M. Abazov et al., Phys. Rev. Lett. 104, 061804 (2010). (Cited on page 6.)
[59] T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956). (Cited on pages 7, 9,
and 10.)
[60] C. S. Wu et al., Phys. Rev. 105, 1413 (1957). (Cited on page 7.)
[61] E. Fermi, Zeitschrift für Physik 88, 161 (1934). (Cited on page 7.)
[62] G. Gamow and E. Teller, Phys. Rev. 49, 895 (1936). (Cited on pages 7 and 10.)
[63] G. Sudarshan and R. Marshak, Phys. Rev. 109, 1860 (1958). (Cited on page 7.)
[64] R. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). (Cited on page 7.)
[65] J. D. Jackson, S. B. Treiman, and H. W. Wyld, Nucl. Phys. 4, 206 (1957).
(Cited on pages 10 and 11.)
[66] H. P. Mumm et al., (2011), arXiv:1104.2778v2 [nucl-ex]. (Cited on page 11.)
[67] T. Soldner et al., Phys. Lett. B 581, 49 (2004). (Cited on page 11.)
[68] A. Kozela et al., Phys. Rev. Lett. 102, 172301 (2009). (Cited on pages 11 and 13.)
[69] A. Kozela et al., Acta Physica Polonica B 42, 789 (2011). (Cited on pages 11
and 13.)
[70] D. H. Wilkinson, Nucl. Phys. A 377, 474 (1982). (Cited on pages 13, 16, 17,
and 31.)
BIBLIOGRAPHY 237
[71] I. S. Towner and J. C. Hardy, Phys. Rev. C 77, 025501 (2008). (Cited on
page 13.)
[72] H. Abele et al., Eur. Phys. J. C 33, 1 (2004). (Cited on page 13.)
[73] A. Sirlin, Phys. Rev. 164, 1767 (1967). (Cited on pages 13 and 17.)
[74] F. Glück, Phys. Rev. D 47, 2840 (1993). (Cited on pages 13, 15, 16, 17, 31,
and 165.)
[75] I. S. Towner and J. C. Hardy, Rev. Part. Phys. 73, 046301 (2010). (Cited on
pages 13, 25, and 31.)
[76] J. C. Hardy and I. S. Towner, Phys. Rev. C 79, 055502 (2009). (Cited on
pages 13, 20, 22, 25, 28, 30, 31, and 32.)
[77] O. Nachtmann, Z. Phys. 215, 505 (1968),
please note a correction of a sign error in Eq. (4.5), found by C. Habeck, PhD Thesis,
University of Sussex, 1997. (Cited on page 14.)
[78] C. Habeck, PhD thesis, University of Sussex, 1997. (Cited on page 14.)
[79] P. G. Dawber et al., Nucl. Instr. and Meth. A 440, 543 (2000). (Cited on page 14.)
[80] S. Baeßler, Die Betaasymmetrie im Zerfall des freien Neutrons, PhD thesis,
Ruprecht-Karls-Universität Heidelberg, 1996. (Cited on page 16.)
[81] B. Povh et al., Teilchen und Kerne, Springer, Berlin, Heidelberg, 2004. (Cited on
page 16.)
[82] J. Pati and A. Salam, Phys. Rev. D 10, 275 (1974). (Cited on page 21.)
[83] R. Mohapatra and J. Pati, Phys. Rev. D 11, 566 (1975). (Cited on page 21.)
[84] M. A. Bég, Phys. Rev. Lett. 38, 1252 (1977). (Cited on page 21.)
[85] J. Döhner, Der Beta-Zerfall freier Neutronen und rechtshändige Ströme, PhD
thesis, Ruprecht-Karls-Universität Heidelberg, 1990. (Cited on page 21.)
[86] M. Schumann, Measurement of Neutrino and Proton Asymmetry in the Decay
of polarized Neutrons, PhD thesis, Ruperto-Carola University of Heidelberg, 2007.
(Cited on pages 21 and 211.)
[87] Y. Zhang et al., Phys. Rev. D 76, 091301 (2007). (Cited on pages 21 and 30.)
[88] C. A. Baker and other, Phys. Rev. Lett. 97, 131801 (2006). (Cited on page 21.)
[89] W. T. Eadie et al., Statistical Methods in Experimental Physics, North-Holland
Publishing Co., 1971. (Cited on page 22.)
[90] W. H. Press et al., Numerical Recipes: The Art of Scientific Computing, Cam-
bridge University Press, 3rd edition, 2007, Sec. 15.6. (Cited on pages 22 and 86.)
[91] V. F. Ezhov et al., Nucl. Instr. and Meth. A 611, 167 (2009). (Cited on pages 23
and 24.)
238 BIBLIOGRAPHY
[92] V. F. Ezhov, Neutron lifetime measuring using magnetic trap, in Proceedings
of the 7th International UCN Workshop “Ultracold and Cold Neutrons. Physics and
Sources.”, 2009. (Cited on page 23.)
[93] A. P. Serebrov et al., Phys. Lett. B 605, 72 (2005). (Cited on pages 23 and 24.)
[94] A. Pichlmaier et al., Phys. Lett. B 693, 221 (2010). (Cited on page 23.)
[95] C. Amsler et al., Phys. Lett. B 667, 1 (2008), and 2009 partial update for the
2010 edition. (Cited on page 23.)
[96] R. W. Pattie et al., Phys. Rev. Lett. 102, 01231 (2009). (Cited on page 23.)
[97] H. Abele, private communication, 2010. (Cited on page 23.)
[98] F. E. Wietfeldt et al., Nucl. Instr. and Meth. A 611, 207 (2009). (Cited on
pages 23, 39, and 40.)
[99] D. Počanić et al., Nucl. Instr. and Meth. A 611, 211 (2009). (Cited on pages 23,
39, 41, and 231.)
[100] K. P. Hickerson, The Fierz Interference Term in Beta Decay Spectrum of Ultra-
cold Neutrons, in UCN Workshop, November 6-7 2009, Santa Fe, New Mexico, 2009,
http://neutron.physics.ncsu.edu/UCN_Workshop_09/Hickerson_SantaFe_2009.pdf.
(Cited on page 23.)
[101] B. Märkisch et al., Nucl. Instr. and Meth. A 611, 216 (2009). (Cited on page 23.)
[102] B. Plaster et al., Nucl. Instr. and Meth. A 595, 587 (2008). (Cited on page 23.)
[103] R. Alarcon et al., The abBA Experiment proposal, 2007, http://nab.phys.
virginia.edu/ABBA_proposal_2007.pdf. (Cited on pages 23 and 24.)
[104] D. Dubbers et al., Nucl. Instr. and Meth. A 596, 238 (2008),
for an extended version, see arXiv:0709.4440v1 [nucl-ex]. (Cited on pages 23, 211,
212, 229, and 230.)
[105] W. S. Wilburn et al., Rev. Mex. Fís. 55, 119 (2009). (Cited on page 24.)
[106] R. Alarcon et al., PANDA @ SNS proposal, 2008, http://research.physics.
lsa.umich.edu/chupp/panda/PANDA@SNS.pdf. (Cited on page 24.)
[107] M. Dewey et al., Nucl. Instr. and Meth. A 611, 189 (2009). (Cited on page 24.)
[108] H. M. Shimizu, in Proceedings of the 7th International UCN Workshop, Saint
Petersburg, Russia, 2009, http://cns.pnpi.spb.ru/ucn/articles/Shimizu.pdf.
(Cited on page 24.)
[109] S. Arzumanov et al., Nucl. Instr. and Meth. A 611, 186 (2009). (Cited on page 24.)
[110] A. P. Serebrov, in Proceedings of the 7th International UCN Workshop, Saint Pe-
tersburg, Russia, 2009, http://cns.pnpi.spb.ru/ucn/articles/Serebrov3.pdf.
(Cited on page 24.)
BIBLIOGRAPHY 239
[111] P. L. Walstron et al., Nucl. Instr. and Meth. A 599, 82 (2009). (Cited on
page 24.)
[112] S. Materne et al., Nucl. Instr. and Meth. A 611, 176 (2009).
[113] K. K. H. Leung and O. Zimmer, Nucl. Instr. and Meth. A 611, 181 (2009).
[114] C. M. O’Shaughnessy et al., Nucl. Instr. and Meth. A 611, 171 (2009). (Cited
on page 24.)
[115] Y. A. Mostovoĭ and A. Frank, J. Exp. and Theor. Phys. Lett. 24, 38 (1976),
Translated from. (Cited on page 24.)
[116] J. C. Hardy and I. S. Towner, Phys. Rev. C 71, 055501 (2005). (Cited on
pages 25 and 31.)
[117] G. A. Miller and A. Schwenk, Phys. Rev. C 78, 035501 (2008). (Cited on
page 25.)
[118] G. A. Miller and A. Schwenk, arXiv:0910.2790, 2009.
[119] N. Auerbach, Phys. Rev. C 79, 035502 (2009).
[120] H. Liang, N. Van Giai, and J. Meng, Phys. Rev. C 79, 064316 (2009). (Cited
on page 25.)
[121] O. Naviliat-Čunčić and N. Severijns, Phys. Rev. Lett. 102, 142302 (2009).
(Cited on page 26.)
[122] D. Počanić et al., Phys. Rev. Lett. 93, 181803 (2004). (Cited on page 26.)
[123] A. P. Serebrov et al., Zh. Éksp. Teor. Fiz. 113, 1963 (1998). (Cited on pages 26
and 27.)
[124] A. P. Serebrov et al., J. Exp. and Theor. Phys. 86, 1074 (1998). (Cited on
pages 26 and 27.)
[125] M. Schumann et al., Phys. Rev. Lett. 99, 191803 (2007). (Cited on page 27.)
[126] A. I. Boothroyd, J. Markey, and P. Vogel, Phys. Rev. C 29, 603 (1984).
(Cited on page 28.)
[127] J. Camps, Search for right-handed currents and tensor type interactions in the beta
decay of polarised nuclei, PhD thesis, Kath. Univ. Leuven, 1997. (Cited on pages 28
and 30.)
[128] N. Severijns et al., Hyperfine Interact. 129, 223 (2000). (Cited on pages 28
and 30.)
[129] F. Wauters et al., Phys. Rev. C 80, 062501 (2009), arXiv:0901.0081v2 [nucl-ex].
(Cited on page 28.)
[130] F. Wauters et al., Phys. Rev. C 82, 055502 (2010), arXiv:1005.5034v1 [nucl-ex].
(Cited on page 28.)
240 BIBLIOGRAPHY
[131] S. Profumo, M. J. Ramsey-Musolf, and S. Tulin, Phys. Rev. D 75, 075017
(2007). (Cited on pages 28 and 34.)
[132] M. Schumann, Precision measurements in neutron decay, in Proceedings of the
XLIInd Rencontres de Moriond - Electroweak Interactions and Unified Theories,
March 10-17 2007, La Thuile, Italy, 2007, arXiv:0705.3769v2[hep-ph]. (Cited on
page 28.)
[133] G. Barenboim et al., Phys. Rev. D 55, 4213 (1997). (Cited on page 30.)
[134] R. P. MacDonald et al., Phys. Rev. D 78, 032010 (2008). (Cited on pages 29
and 30.)
[135] M. Czakon, J. Gluza, and M. Zrałek, Phys. Lett. B 458, 355 (1999). (Cited
on page 30.)
[136] V. M. Abazov et al., Phys. Rev. Lett. 100, 031804 (2008). (Cited on pages 29
and 30.)
[137] N. Severijns et al., Nucl. Phys. A 629, 429c (1998). (Cited on page 29.)
[138] N. Severijns et al., Phys. Rev. C 78, 055501 (2008). (Cited on page 31.)
[139] J. C. Hardy and I. S. Towner, Nucl. Phys. A 254, 221 (1975). (Cited on
page 32.)
[140] H. Klapdor-Kleingrothaus, I. Krivosheina, and R. Viollier, editors,
Physics Beyond the Standard Model of Particles, Cosmology and Astrophysics: Pro-
ceedings of the Fifth International Conference Beyond 2010, World Scientific, Sin-
gapore, 2011. (Cited on page 33.)
[141] C. A. Gagliardi, R. Tribble, and N. J. Williams, Phys. Rev. D 72, 073002
(2005). (Cited on page 34.)
[142] V. Cirigliano, private communication to D. Počanić, 2009. (Cited on page 34.)
[143] M. Bychkov et al., Phys. Rev. Lett. 103, 051802 (2009). (Cited on page 34.)
[144] P. Herczeg, private communication to D. Počanić, 2004. (Cited on page 34.)
[145] http://pibeta.phys.virginia.edu/. (Cited on page 34.)
[146] http://pen.phys.virginia.edu/. (Cited on page 34.)
[147] M. Doi, T. Kotani, and E. Takasugi, Prog. Theor. Phys. Suppl. 83, 1 (1985).
(Cited on pages 34 and 35.)
[148] H. Päs et al., Phys. Lett. B 453, 194 (1999). (Cited on pages 34 and 35.)
[149] R. Mohapatra and A. Y. Smirnov, Annu. Rev. Nucl. Part. Sci. 56, 569 (2006).
(Cited on page 34.)
[150] J. Hirsch, H. V. Klapdor-Kleingrothaus, and O. Panella, Phys. Lett. B
374, 7 (1996). (Cited on page 35.)
BIBLIOGRAPHY 241
[151] H. V. Klapdor-Kleingrothaus and I. V. Krivosheina, Modern Phys. Lett. A
21, 1547 (2006). (Cited on page 35.)
[152] T. M. Ito andG. Prezeau, Phys. Rev. Lett. 94, 161802 (2005). (Cited on page 35.)
[153] V. M. Lobashev et al., Nucl. Phys. A 719, C153 (2003). (Cited on pages 35
and 47.)
[154] C. Kraus et al., Eur. Phys. J. C 40, 447 (2005). (Cited on pages 35 and 47.)
[155] J. Angrik et al., FZKA Scientific Report 7090 (2004), http://bibliothek.fzk.
de/zb/berichte/FZKA7090.pdf. (Cited on pages 35 and 47.)
[156] E. Komatsu et al., Astrophys. J. Suppl. Ser. 180, 330 (2009). (Cited on page 35.)
[157] M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). (Cited on page 35.)
[158] V. K. Grigor’ev et al., Sov. J. Nucl. Phys. 6, 239 (1968). (Cited on page 36.)
[159] J. Byrne, J. Res. Natl. Inst. Stand. Technol. 110, 395 (2005). (Cited on page 37.)
[160] J. Byrne et al., Europhys. Lett. 33, 187 (1996). (Cited on page 37.)
[161] Y. A. Mostovoĭ et al., Phys. Atomic Nucl. 64, 1955 (2001), Translated from YAF
64 2040. (Cited on page 39.)
[162] F. E. Wietfeldt et al., Nucl. Instr. and Meth. A 545, 181 (2005). (Cited on
page 39.)
[163] R. Alarcon et al., Nab proposal update and funding request, 2010, http://nab.
phys.virginia.edu/nab_doe_fund_prop.pdf. (Cited on pages 39, 41, and 231.)
[164] B. Yerozolimsky et al., (2004), arXiv:nucl-ex/0401014. (Cited on page 39.)
[165] S. Balashov and Y. Mostovoy, (1994), Preprint IAE-5718/2, Russian Research
Center Kurchatov Institute, Moscow. (Cited on page 39.)
[166] G. Jones, aCORN: ‘a’ CORrelation in Neutron Decay, in Proceedings of the 8th
International UCN Workshop “Ultracold and Cold Neutrons. Physics and Sources.”,
2011. (Cited on page 41.)
[167] J. Bowman, J. Res. Natl. Inst. Stand. Technol. 110, 407 (2005). (Cited on page 41.)
[168] F. Ayala Guardia, First Tests of the neutron decay spectrometer aSPECT,
Master’s thesis, Johannes Gutenberg-Universität Mainz, 2005. (Cited on pages 43,
62, and 89.)
[169] P. Kruit and F. H. Read, J. Phys. E 16, 313 (1983). (Cited on pages 45 and 47.)
[170] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, 3rd edition, 1975.
(Cited on pages 46, 214, and 218.)
[171] T. Hsu and J. L. Hirshfield, Rev. Sci. Instrum. 47, 236 (1976). (Cited on
page 47.)
242 BIBLIOGRAPHY
[172] G. Beamson, H. Q. Porter, and D. W. Turner, J. Phys. E 13, 64 (1980).
(Cited on page 47.)
[173] V. M. Lobashev and P. E. Spivak, Nucl. Instr. and Meth. A 240, 305 (1985).
(Cited on page 47.)
[174] A. Picard et al., Nucl. Instr. and Meth. B 63, 345 (1992). (Cited on pages 47
and 72.)
[175] M. Beck et al., Eur. Phys. J. A 47, 45 (2011). (Cited on page 47.)
[176] T. G. Northrop, The Adiabatic Motion of Charged Particles, Interscience Publ.,
1963. (Cited on pages 47 and 73.)
[177] R. Dendy, editor, Plasma Physics: An Introductional Course, Cambridge Univer-
sity Press, 1993. (Cited on pages 47 and 73.)
[178] F. M. Penning, Physica (Utrecht) 3, 873 (1936). (Cited on page 56.)
[179] H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967). (Cited on page 56.)
[180] B. Müller, Umbau des Mainzer Neutrinomassenxperiments und Untergrundun-
tersuchungen im Hinblick auf KATRIN, Master’s thesis, Johannes Gutenberg-
Universität Mainz, 2002. (Cited on page 57.)
[181] F. M. Fränkle, Erste Messungen der elektromagnetischen Eigenschaften des KA-
TRIN Vorspektrometers, Master’s thesis, Universität Karlsruhe (IEKP), 2006.
[182] K. A. Hugenberg, Design of the electrode system for the KATRIN main spec-
trometer, Master’s thesis, Westfälische Wilhelms-Universität Münster, 2008.
[183] F. Habermehl for the KATRIN Collaboration, J. Phys.: Conf. Series 120,
052301 (2008). (Cited on page 57.)
[184] K. Valerius, Spectrometer-related background processes and their suppression in
the KATRIN experiment, PhD thesis, Westfälische Wilhelms-Universität Münster,
2009. (Cited on page 57.)
[185] F. M. Fränkle, Background Investigations of the KATRIN Pre-Spectrometer, PhD
thesis, KIT/IEKP, 2010. (Cited on page 57.)
[186] F. Glück, The Penning discharge, to be published. (Cited on page 57.)
[187] J. Erhart, private communication, 2011. (Cited on page 61.)
[188] F. Glück, Axisymmetric electric field calculation with zonal harmonic expansion,
to be published in PIER. (Cited on page 61.)
[189] F. Glück, Axisymmetric magnetic field calculation with zonal harmonic expansion,
to be published in PIER.
[190] F. Glück, Relativistic charged particle tracking with 8th order Runge-Kutta method,
to be published in PIER. (Cited on page 61.)
BIBLIOGRAPHY 243
[191] F. Glück, Axisymmetric electric field calculation with BEM and with
zonal harmonic expansion, http://project8.physics.ucsb.edu/twiki/pub/
Main/KatrinEgun/BoundaryElementMethod.pdf, 2007. (Cited on page 61.)
[192] T. J. Corona, Tools for Electromagnetic Field Simulation in the KATRIN Exper-
iment, Master’s thesis, Massachusetts Institute of Technology, 2009.
[193] F. Glück, Axisymmetric electric field calculation with charge density method, to
be published.
[194] F. Glück, Axisymmetric magnetic field calculation with Legendre polynomials and
elliptic integrals, to be published.
[195] F. Glück, Axisymmetric coil design for homogeneous magnetic field, to be pub-
lished. (Cited on page 61.)
[196] http://www.comsol.com/. (Cited on page 62.)
[197] G. Petzoldt, Measurement of the electron-antineutrino angular correlation coeffi-
cient a in neutron beta decay with the spectrometer aSPECT, PhD thesis, Technische
Universität München, 2007. (Cited on pages 64, 70, 96, 106, and 129.)
[198] PNSensor GmbH, http://www.pnsensor.de. (Cited on page 65.)
[199] G. Lutz, Semiconductor Radiation Detectors, Springer, Berlin, Heidelberg, New
York, 1st edition, 2007, http://dx.doi.org/10.1007/978-3-540-71679-2. (Cited
on page 65.)
[200] G. F. Knoll, Radiation detection and measurement, Wiley, Hoboken, 4th edition,
2010. (Cited on page 65.)
[201] E. Gatti and P. Rehak, Nucl. Instr. and Meth. A 225, 608 (1984). (Cited on
page 65.)
[202] R. Brun and F. Rademakers, Nucl. Instr. and Meth. A 389, 81 (1997), http:
//root.cern.ch. (Cited on page 70.)
[203] H. Backe et al., Phys. Scr. T. 22, 98 (1988). (Cited on page 72.)
[204] C. Weinheimer et al., Phys. Lett. B 460, 219 (1999). (Cited on page 72.)
[205] R. Dobrozemsky, Nucl. Instr. and Meth. 118, 1 (1974). (Cited on pages 72, 167,
174, and 175.)
[206] C. F. Giese andW. R. Gentry, Phys. Rev. A 10, 2156 (1974). (Cited on page 74.)
[207] R. Schinke, Chem. Phys. 24, 379 (1977).
[208] H. Krüger and R. Schrinke, J. Chem. Phys. 66, 5087 (1977).
[209] G. Niedner et al., J. Chem. Phys. 87, 2685 (1987).
[210] D. Dhuicq and C. Benoit, J. Phys. B 24, 3599 (1991). (Cited on page 74.)
244 BIBLIOGRAPHY
[211] R. Muñoz Horta, in Proceedings of the XIV International Seminar on Interaction
of Neutrons with Nuclei, Dubna, 2007, 2007. (Cited on page 75.)
[212] J. F. Ziegler et al., SRIM - The Stopping and Range of Ions in Matter. Computer
Program, http://www.srim.org, 2008. (Cited on page 79.)
[213] H. Häse et al., Nucl. Instr. and Meth. A 458, 453 (2002). (Cited on page 82.)
[214] H. Abele et al., Nucl. Instr. and Meth. A 562, 407 (2006). (Cited on page 83.)
[215] http://www.ill.eu/instruments-support/instruments-groups/instruments/
pf1b/characteristics/. (Cited on page 83.)
[216] B. Singh, Nucl. Data Sheets 108, 197 (2007). (Cited on page 85.)
[217] E. Browne and J. K. Tuli, Nucl. Data Sheets 111, 1093 (2010). (Cited on
page 85.)
[218] J. Krempel, Optimierung und Durchführung einer Beta-Asymmetriemessung im
Zerfall polarisierter Neutronen, Master’s thesis, Ruprecht-Karls-Universität Heidel-
berg, 2004. (Cited on page 85.)
[219] Group3 Technology Ltd., http://www.group3technology.com/dlarea/docs/
MPT-141%20spec.pdf. (Cited on page 88.)
[220] S. J. B. Reed and N. G. Ware, J. Phys. E 5, 582 (1972). (Cited on page 92.)
[221] J. A. Bearden, Rev. Mod. Phys. 39, 78 (1967). (Cited on pages 91 and 92.)
[222] R. G. Helmer and C. van der Leun, Nucl. Instr. and Meth. A 450, 35 (2000).
(Cited on page 91.)
[223] W. R. Leo, Techniques for Nuclear and Particle Physics Experiments, Springer-
Verlag, Berlin, Heidelberg, New York, 2nd edition, 1994. (Cited on page 100.)
[224] F. Glück, private communication, 2010. (Cited on page 114.)
[225] M. Simson, private communication, 2010. (Cited on pages 122 and 123.)
[226] D. Drouin et al., CASINO - monte CArlo SImulation of electroN trajectory
in sOlids, Computer program. http://www.gel.usherbrooke.ca/casino, 2001,
please note that the new 3D version v3.2 is available since August 05, 2011. (Cited
on page 122.)
[227] D. Drouin et al., Scanning 29, 92 (2007). (Cited on pages 122 and 131.)
[228] P. Hovington et al., Scanning 19, 1 (1997), 10.1002/sca.4950190101. (Cited
on pages 122 and 131.)
[229] F. E. Wietfeldt et al., Nucl. Instr. and Meth. A 538, 574 (2005). (Cited on
pages 131, 132, and 133.)
[230] S. M. Seltzer, Int. J. Appl. Radiat. Isot. 42, 917 (1991). (Cited on page 131.)
BIBLIOGRAPHY 245
[231] GEANT4 - a toolkit for the simulation of the passage of particles through matter,
Computer program. http://geant4.cern.ch/. (Cited on page 131.)
[232] S. Agostinelli et al., Nucl. Instr. and Meth. A 506, 250 (2003). (Cited on
page 131.)
[233] J. Allison et al., IEEE Transactions on Nuclear Science 53, 270 (2006). (Cited
on page 131.)
[234] H. Kanter, Annalen der Physik 20, 144 (1957). (Cited on page 134.)
[235] F. Glück, private communication, 2008. (Cited on page 138.)
[236] B. Märkisch, private communication to T. Soldner, 2010. (Cited on page 141.)
[237] A. R. Müller et al., Nucl. Instr. and Meth. A 582, 395 (2007). (Cited on page 159.)
[238] A. R. Müller, Characterization of solid deuterium as a source material for ul-
tracold neutrons (UCN) and development of a detector concept for the detection of
protons from the neutron decay, PhD thesis, Technische Universität München, 2008.
(Cited on page 159.)
[239] I. Baikie and P. Estrup, Rev. Sci. Instrum. 69, 3902 (1998). (Cited on page 167.)
[240] N. A. Robertson et al., Class. Quantum Grav. 23, 2665 (2006). (Cited on
pages 176, 178, 181, 190, and 206.)
[241] S. E. Pollack, S. Schlamminger, and J. H. Gundlach, Phys. Rev. Lett. 101,
071101 (2008). (Cited on page 167.)
[242] R. Hodges, Application of a Scanning Kelvin Probe to Measure Local Surface
Potential Inhomogeneities, Bachelor’s thesis, University of Virginia, 2009. (Cited on
pages 167, 168, 170, 174, 175, and 176.)
[243] S. McGovern, On the Determination of Surface Potential Variations with a Kelvin
Probe, Master’s thesis, University of Virginia, 2010. (Cited on pages 167, 170, 174,
175, 176, 177, 178, 181, 182, 183, 191, and 231.)
[244] W. M. Haynes, editor, CRC Handbook of Chemistry and Physics, CRC Press Inc.,
91st edition, 2010. (Cited on page 169.)
[245] F. Rossi and G. I. Opat, 25, 1349 (1992). (Cited on page 168.)
[246] D. Sauer and M. Stuve, Changes in Work Function of Pt(111), http://www.
mcallister.com/papers/stuve.html., 1994. (Cited on page 168.)
[247] J. Camp, T. W. Darling, and R. E. Brown, J. Appl. Phys. 71, 783 (1992).
(Cited on pages 168 and 187.)
[248] KP Technology Ltd., Ambient Scanning Kelvin Probe Manual. (Cited on
pages 170, 173, and 176.)
[249] I. D. Baikie, Old Principles, New Techniques: A Novel UHV Kelvin Probe and
its Application in the Study of Semiconductor Surfaces., PhD thesis, University of
Twente, 1988, ISBN 90-9002444-1. (Cited on pages 170 and 173.)
246 BIBLIOGRAPHY
[250] http://www.kelvinprobe.info/technique-pccontrol.htm. (Cited on page 170.)
[251] I. Baikie, Rev. Sci. Instrum. 62, 1326 (1991). (Cited on page 173.)
[252] G.-N. Luo, K. Yamaguchi, T. Terai, and M. Yamawaki, Rev. Sci. Instrum.
72, 2350 (2001). (Cited on page 174.)
[253] I. Baikie, private communication to R. Hodges, 2008. (Cited on page 174.)
[254] W. Li and D. Y. Li, J. Chem. Phys. 122, 064708 (2005). (Cited on page 175.)
[255] M. Dressend, private communication, 2010. (Cited on page 176.)
[256] H. Diettrich, private communication, 2010. (Cited on page 176.)
[257] S. Baeßler, private communication, 2011. (Cited on pages 176, 178, 184, and 207.)
[258] I. Baikie, private communication to S. McGovern, 2009. (Cited on page 177.)
[259] S. McGovern, private communication, 2010. (Cited on page 178.)
[260] M. Baumgärtner and C. J. Raub, Platinum Metals Rev. 29, 155 (1985). (Cited
on pages 184, 187, and 190.)
[261] http://www.universalmeasurement.com/MahrFederalFinish.pdf. (Cited on
page 187.)
[262] M. Zanker, private communication, 2010. (Cited on page 189.)
[263] S. L. Firebaugh, K. F. Jensen, and M. A. Schmidt, JMEMS 7, 128 (1998).
(Cited on page 189.)
[264] L. L. Melo et al., Journal of Metastable and Nanocrystalline Materials 20-21, 623
(2004). (Cited on page 189.)
[265] R. Petrović et al., J. Sci. Instrum. 66, 483 (2001). (Cited on page 189.)
[266] A. W. Groenland, Degradation of processes of platinum thin films on a silicon
nitride surface, Bachelor’s thesis, University of Twente, 2004. (Cited on page 189.)
[267] W. Chen et al., Appl. Phys. Lett. 97, 211107 (2010). (Cited on page 189.)
[268] M. Beck, private communication, 2011. (Cited on page 207.)
[269] M. Schumann et al., Phys. Rev. Lett. 100, 151801 (2008), arXiv:0712.2442 [hep-
ph]. (Cited on page 211.)
[270] M. Horvath, private communication, 2011. (Cited on page 212.)
[271] C. Gösselsberger et al., Physics Procedia (2011), accepted,
see also: Badurek G, Gösselsberger C and Jericha E, Physica B 406 2458 (2011).
(Cited on page 230.)
[272] C. Gösselsberger et al., (2011), submitted to J. Phys.: Conf. Series.
[273] C. Klauser et al., (2011), submitted to J. Phys.: Conf. Series. (Cited on page 230.)
[274] A. Senft, Calculations on Proton Transmission for PERC, Project work, Technis-
che Universität München, 2010. (Cited on page 231.)
Erklärung
Hiermit versichere ich, dass ich die vorliegende Arbeit selbstständig verfasst und keine
anderen als die von mir angegebenen Quellen und Hilfsmittel verwendet habe.
Die Arbeit ist in dieser oder ähnlicher Form noch nicht als Prüfungsarbeit eingereicht
worden.
Mainz, im August 2011
Gertrud Emilie Konrad
247

Acknowledgments
I would like to take this opportunity to express my heartfelt thanks to all those who have
contributed to the success of this work.
249

Curriculum Vitæ
Birthday April 2nd, 1977
Birthplace Worms, Germany
Since 01/2011 Project assistant
at the Institute of Atomic and Subatomic Physics
of the Vienna University of Technology
Since 01/2006 PhD student
at the Johannes Gutenberg-University of Mainz
2003 - 2005 Student in physics
at the Johannes Gutenberg-University of Mainz
10/2003 Diploma in mathematics
at the Johannes Gutenberg-University of Mainz
Title: “Eigenwertcharakterisierung und Lösungs-
geometrie bei semilinearen gewöhnlichen Differential-
gleichungen höherer Ordnung” (“Characterization
of eigenvalues and geometry of solutions to semi-
linear differential equations of higher order”)
10/1999 Prediploma in mathematics
at the Johannes Gutenberg-University of Mainz
1996 - 2003 Student in mathematics, physics and computer science
at the Johannes Gutenberg-University of Mainz
06/1996 Abitur at the Eleonoren-Gymnasium of Worms
1987 - 1996 High school Eleonoren-Gymnasium of Worms
1983 - 1987 Primary school Otto Hahn-Schule of Westhofen
251